The Algebra and Geometry of VectorsMath 218

Brian D. Fitzpatrick

Duke University

November 1, 2019

MATH

Overview

MotivationWhat We KnowWhat We Want

Vectors in R2

Geometric InterpretationTails and TipsCoordinatesLength

Vectors in R3

Geometric InterpretationLength

Vectors in Rn

“Geometric” InterpretationLength

Scalar-Vector MultiplicationGeometric InterpretationAlgebraic ComparisonUnit VectorsNormalization

Vector AdditionGeometric InterpretationProperties

The Dot ProductDefinitionPropertiesLengthsAngles

MotivationWhat We Know

So far, the vector operations we have learned are

scalar-vector muliplication c · #»v vector addition #»v + #»w

These operations are algebraic

c ·

v1...vn

=

c · v1...

c · vn

v1

...vn

+

w1...wn

=

v1 + w1...

vn + wn

MotivationWhat We Know

So far, the vector operations we have learned are

scalar-vector muliplication c · #»v vector addition #»v + #»w

These operations are algebraic

c ·

v1...vn

=

c · v1...

c · vn

v1

...vn

+

w1...wn

=

v1 + w1...

vn + wn

MotivationWhat We Know

So far, the vector operations we have learned are

scalar-vector muliplication c · #»v vector addition #»v + #»w

These operations are algebraic

c ·

v1...vn

=

c · v1...

c · vn

v1...vn

+

w1...wn

=

v1 + w1...

vn + wn

MotivationWhat We Know

So far, the vector operations we have learned are

scalar-vector muliplication c · #»v vector addition #»v + #»w

These operations are algebraic

c ·

v1...vn

=

c · v1...

c · vn

v1

...vn

+

w1...wn

=

v1 + w1...

vn + wn

MotivationWhat We Want

Can we interpret these operations geometrically?

Vectors in R2

Geometric Interpretation

Geometrically, a vector in R2 is represented by an arrow in thexy -plane.

x

y

#»a

#»

b

#»v

#»w

#»x#»y

#»z

Vectors in R2

Geometric Interpretation

Geometrically, a vector in R2 is represented by an arrow in thexy -plane.

x

y

#»a

#»

b

#»v

#»w

#»x#»y

#»z

Vectors in R2

Geometric Interpretation

Geometrically, a vector in R2 is represented by an arrow in thexy -plane.

x

y

#»a

#»

b

#»v

#»w

#»x#»y

#»z

Vectors in R2

Geometric Interpretation

Geometrically, a vector in R2 is represented by an arrow in thexy -plane.

x

y

#»a

#»

b

#»v

#»w

#»x#»y

#»z

Vectors in R2

Geometric Interpretation

Geometrically, a vector in R2 is represented by an arrow in thexy -plane.

x

y

#»a

#»

b

#»v

#»w

#»x#»y

#»z

Vectors in R2

Geometric Interpretation

Geometrically, a vector in R2 is represented by an arrow in thexy -plane.

x

y

#»a

#»

b

#»v

#»w

#»x#»y

#»z

Vectors in R2

Geometric Interpretation

Geometrically, a vector in R2 is represented by an arrow in thexy -plane.

x

y

#»a

#»

b

#»v

#»w

#»x

#»y

#»z

Vectors in R2

Geometric Interpretation

Geometrically, a vector in R2 is represented by an arrow in thexy -plane.

x

y

#»a

#»

b

#»v

#»w

#»x#»y

#»z

Vectors in R2

Geometric Interpretation

Geometrically, a vector in R2 is represented by an arrow in thexy -plane.

x

y

#»a

#»

b

#»v

#»w

#»x#»y

#»z

Vectors in R2

Tails and Tips

Every arrow emanates from a tail and terminates at a tip.

x

y

−3 −2 −1 1 2 3

−2

−1

1

2

#»v

=# »

PQ

•P(−3,−1)

“tail”

•Q(2, 2)“tip”

#»w

=# »

RS

•R(−2,−2)

“tail”

•S(3,−1)

“tip”

Vectors in R2

Tails and Tips

Every arrow emanates from a tail and terminates at a tip.

x

y

−3 −2 −1 1 2 3

−2

−1

1

2

#»v

=# »

PQ

•P(−3,−1)

“tail”

•Q(2, 2)“tip”

#»w

=# »

RS

•R(−2,−2)

“tail”

•S(3,−1)

“tip”

Vectors in R2

Tails and Tips

Every arrow emanates from a tail and terminates at a tip.

x

y

−3 −2 −1 1 2 3

−2

−1

1

2

#»v

=# »

PQ

•P(−3,−1)

“tail”

•Q(2, 2)“tip”

#»w

=# »

RS

•R(−2,−2)

“tail”

•S(3,−1)

“tip”

Vectors in R2

Tails and Tips

Every arrow emanates from a tail and terminates at a tip.

x

y

−3 −2 −1 1 2 3

−2

−1

1

2

#»v

=# »

PQ

•P(−3,−1)

“tail”

•Q(2, 2)“tip”

#»w

=# »

RS

•R(−2,−2)

“tail”

•S(3,−1)

“tip”

Vectors in R2

Tails and Tips

Every arrow emanates from a tail and terminates at a tip.

x

y

−3 −2 −1 1 2 3

−2

−1

1

2

#»v

=# »

PQ

•P(−3,−1)

“tail”

•Q(2, 2)“tip”

#»w

=# »

RS

•R(−2,−2)

“tail”

•S(3,−1)

“tip”

Vectors in R2

Tails and Tips

Every arrow emanates from a tail and terminates at a tip.

x

y

−3 −2 −1 1 2 3

−2

−1

1

2

#»v =# »

PQ

•P(−3,−1)

“tail”

•Q(2, 2)“tip”

#»w

=# »

RS

•R(−2,−2)

“tail”

•S(3,−1)

“tip”

Vectors in R2

Tails and Tips

Every arrow emanates from a tail and terminates at a tip.

x

y

−3 −2 −1 1 2 3

−2

−1

1

2

#»v =# »

PQ

•P(−3,−1)

“tail”

•Q(2, 2)“tip”

#»w

=# »

RS

•R(−2,−2)

“tail”

•S(3,−1)

“tip”

Vectors in R2

Tails and Tips

Every arrow emanates from a tail and terminates at a tip.

x

y

−3 −2 −1 1 2 3

−2

−1

1

2

#»v =# »

PQ

•P(−3,−1)

“tail”

•Q(2, 2)“tip”

#»w

=# »

RS

•R(−2,−2)

“tail”

•S(3,−1)

“tip”

Vectors in R2

Tails and Tips

Every arrow emanates from a tail and terminates at a tip.

x

y

−3 −2 −1 1 2 3

−2

−1

1

2

#»v =# »

PQ

•P(−3,−1)

“tail”

•Q(2, 2)“tip”

#»w

=# »

RS

•R(−2,−2)

“tail”

•S(3,−1)

“tip”

Vectors in R2

Tails and Tips

Every arrow emanates from a tail and terminates at a tip.

x

y

−3 −2 −1 1 2 3

−2

−1

1

2

#»v =# »

PQ

•P(−3,−1)

“tail”

•Q(2, 2)“tip”

#»w =# »

RS

•R(−2,−2)

“tail”

•S(3,−1)

“tip”

Vectors in R2

Coordinates

The coordinates of #»v give the “tip to tail” displacement in the x-and y -directions.

v1 =

x1 − x0

v2 =

y1 − y0

#»v=〈v1, v2〉

•

•

(x0, y0)

(x1, y1)

Vectors in R2

Coordinates

The coordinates of #»v give the “tip to tail” displacement in the x-and y -directions.

v1 =

x1 − x0

v2 =

y1 − y0

#»v=〈v1, v2〉

•

•

(x0, y0)

(x1, y1)

Vectors in R2

Coordinates

The coordinates of #»v give the “tip to tail” displacement in the x-and y -directions.

v1 =

x1 − x0

v2 =

y1 − y0

#»v=〈v1, v2〉

•

•

(x0, y0)

(x1, y1)

Vectors in R2

Coordinates

The coordinates of #»v give the “tip to tail” displacement in the x-and y -directions.

v1 = x1 − x0

v2 =

y1 − y0

#»v=〈v1, v2〉

•

•

(x0, y0)

(x1, y1)

Vectors in R2

Coordinates

The coordinates of #»v give the “tip to tail” displacement in the x-and y -directions.

v1 = x1 − x0

v2 = y1 − y0#»v=〈v1, v2〉

•

•

(x0, y0)

(x1, y1)

Vectors in R2

Coordinates

x

y

−3 −2 −1 1 2 3

−2

−1

1

2#»v = 〈2− (−3), 2− (−1)〉 = 〈5, 3〉

•P(−3,−1)

•Q(2, 2)

#»w = 〈3− (−2),−1− (−2)〉 = 〈5, 1〉

•R(−2,−2)

•S(3,−1)

Vectors in R2

Coordinates

Two arrows define the same vector if they have the samecoordinates.

x

y

−3 −2 −1 1 2 3

−2

−1

1

2

#»v =

〈−2, −1〉

#»w =

〈−2, −1〉

#»v = #»w

Vectors in R2

Coordinates

Two arrows define the same vector if they have the samecoordinates.

x

y

−3 −2 −1 1 2 3

−2

−1

1

2

#»v =

〈−2, −1〉

#»w =

〈−2, −1〉

#»v = #»w

Vectors in R2

Coordinates

Two arrows define the same vector if they have the samecoordinates.

x

y

−3 −2 −1 1 2 3

−2

−1

1

2

#»v =

〈−2, −1〉

#»w =

〈−2, −1〉

#»v = #»w

Vectors in R2

Coordinates

Two arrows define the same vector if they have the samecoordinates.

x

y

−3 −2 −1 1 2 3

−2

−1

1

2

#»v =〈−2, −

1〉

#»w =

〈−2, −1〉

#»v = #»w

Vectors in R2

Coordinates

Two arrows define the same vector if they have the samecoordinates.

x

y

−3 −2 −1 1 2 3

−2

−1

1

2

#»v =〈−2, −

1〉

#»w =

〈−2, −1〉

#»v = #»w

Vectors in R2

Coordinates

Two arrows define the same vector if they have the samecoordinates.

x

y

−3 −2 −1 1 2 3

−2

−1

1

2

#»v =〈−2, −

1〉

#»w =〈−2, −

1〉

#»v = #»w

Vectors in R2

Coordinates

Two arrows define the same vector if they have the samecoordinates.

x

y

−3 −2 −1 1 2 3

−2

−1

1

2

#»v =〈−2, −

1〉

#»w =〈−2, −

1〉#»v = #»w

Vectors in R2

Coordinates

Without context, it is convention to plot vectors using the origin asthe tail.

x

y

−3 −2 −1 1 2 3

−2

−1

1

2

#»v = 〈3, 2〉

#»w = 〈−3, 1〉

#»x = 〈−2,−2〉

Vectors in R2

Coordinates

Without context, it is convention to plot vectors using the origin asthe tail.

x

y

−3 −2 −1 1 2 3

−2

−1

1

2

#»v = 〈3, 2〉

#»w = 〈−3, 1〉

#»x = 〈−2,−2〉

Vectors in R2

Coordinates

Without context, it is convention to plot vectors using the origin asthe tail.

x

y

−3 −2 −1 1 2 3

−2

−1

1

2

#»v = 〈3, 2〉

#»w = 〈−3, 1〉

#»x = 〈−2,−2〉

Vectors in R2

Coordinates

Without context, it is convention to plot vectors using the origin asthe tail.

x

y

−3 −2 −1 1 2 3

−2

−1

1

2

#»v = 〈3, 2〉

#»w = 〈−3, 1〉

#»x = 〈−2,−2〉

Vectors in R2

Coordinates

Without context, it is convention to plot vectors using the origin asthe tail.

x

y

−3 −2 −1 1 2 3

−2

−1

1

2

#»v = 〈3, 2〉

#»w = 〈−3, 1〉

#»x = 〈−2,−2〉

Vectors in R2

Length

The length of a vector #»v = 〈v1, v2〉 is ‖ #»v ‖ =

√v21 + v22 .

|v1|

|v2|‖ #»v ‖

Vectors in R2

Length

The length of a vector #»v = 〈v1, v2〉 is ‖ #»v ‖ =

√v21 + v22 .

|v1|

|v2|

‖ #»v ‖

Vectors in R2

Length

The length of a vector #»v = 〈v1, v2〉 is ‖ #»v ‖ =

√v21 + v22 .

|v1|

|v2|‖ #»v ‖

Vectors in R2

Length

The length of a vector #»v = 〈v1, v2〉 is ‖ #»v ‖ =√

v21 + v22 .

|v1|

|v2|‖ #»v ‖

Vectors in R2

Length

Example

The lengths of #»r = 〈1, −1〉 and #»p = 〈3, 2〉 are

‖ #»r ‖ =√

(1)2 + (−1)2 ‖ #»p ‖ =√

(3)2 + (2)2

=√

1 + 1 =√

9 + 4

=√

2 =√

13

Vectors in R2

Length

Example

The lengths of #»r = 〈1, −1〉 and #»p = 〈3, 2〉 are

‖ #»r ‖ =√

(1)2 + (−1)2

‖ #»p ‖ =√

(3)2 + (2)2

=√

1 + 1 =√

9 + 4

=√

2 =√

13

Vectors in R2

Length

Example

The lengths of #»r = 〈1, −1〉 and #»p = 〈3, 2〉 are

‖ #»r ‖ =√

(1)2 + (−1)2

‖ #»p ‖ =√

(3)2 + (2)2

=√

1 + 1

=√

9 + 4

=√

2 =√

13

Vectors in R2

Length

Example

The lengths of #»r = 〈1, −1〉 and #»p = 〈3, 2〉 are

‖ #»r ‖ =√

(1)2 + (−1)2

‖ #»p ‖ =√

(3)2 + (2)2

=√

1 + 1

=√

9 + 4

=√

2

=√

13

Vectors in R2

Length

Example

The lengths of #»r = 〈1, −1〉 and #»p = 〈3, 2〉 are

‖ #»r ‖ =√

(1)2 + (−1)2 ‖ #»p ‖ =√

(3)2 + (2)2

=√

1 + 1

=√

9 + 4

=√

2

=√

13

Vectors in R2

Length

Example

The lengths of #»r = 〈1, −1〉 and #»p = 〈3, 2〉 are

‖ #»r ‖ =√

(1)2 + (−1)2 ‖ #»p ‖ =√

(3)2 + (2)2

=√

1 + 1 =√

9 + 4

=√

2

=√

13

Vectors in R2

Length

Example

The lengths of #»r = 〈1, −1〉 and #»p = 〈3, 2〉 are

‖ #»r ‖ =√

(1)2 + (−1)2 ‖ #»p ‖ =√

(3)2 + (2)2

=√

1 + 1 =√

9 + 4

=√

2 =√

13

Vectors in R2

Length

Other Terms for “Length”

“norm” “magnitude”

Alternate Notation for ‖ #»v ‖Some people write | #»v | instead of ‖ #»v ‖.

Vectors in R2

Length

Other Terms for “Length”

“norm” “magnitude”

Alternate Notation for ‖ #»v ‖Some people write | #»v | instead of ‖ #»v ‖.

Vectors in R3

Geometric Interpretation

We think of vectors in R3 as objects in xyz-space.

x

y

z

3

3

−4

• P(3, 3,−4)

−4

4

5

• Q(−4, 4, 5)

Vectors in R3

Geometric Interpretation

We think of vectors in R3 as objects in xyz-space.

x

y

z

3

3

−4

• P(3, 3,−4)

−4

4

5

• Q(−4, 4, 5)

Vectors in R3

Geometric Interpretation

We think of vectors in R3 as objects in xyz-space.

x

y

z

3

3

−4

• P(3, 3,−4)

−4

4

5

• Q(−4, 4, 5)

Vectors in R3

Geometric Interpretation

Geometrically, a vector in R3 is represented by an arrow inxyz-space.

x

y

z

P(11, 2, 0)

Q(8,−4, 6)

R(−2, 2, 7)

S(1, 8, 1)

#»v

#»w

#»x

#»v =

−2− 82− (−4)

7− 6

=

−1061

#»w =

−2− 112− 27− 0

=

−1307

#»x =

1− 118− 21− 0

=

−1061

Note that #»v = #»x !

Vectors in R3

Geometric Interpretation

Geometrically, a vector in R3 is represented by an arrow inxyz-space.

x

y

z

P(11, 2, 0)

Q(8,−4, 6)

R(−2, 2, 7)

S(1, 8, 1)

#»v

#»w

#»x

#»v =

−2− 82− (−4)

7− 6

=

−1061

#»w =

−2− 112− 27− 0

=

−1307

#»x =

1− 118− 21− 0

=

−1061

Note that #»v = #»x !

Vectors in R3

Geometric Interpretation

Geometrically, a vector in R3 is represented by an arrow inxyz-space.

x

y

z

P(11, 2, 0)

Q(8,−4, 6)

R(−2, 2, 7)

S(1, 8, 1)

#»v

#»w

#»x

#»v =

−2− 82− (−4)

7− 6

=

−1061

#»w =

−2− 112− 27− 0

=

−1307

#»x =

1− 118− 21− 0

=

−1061

Note that #»v = #»x !

Vectors in R3

Geometric Interpretation

Geometrically, a vector in R3 is represented by an arrow inxyz-space.

x

y

z

P(11, 2, 0)

Q(8,−4, 6)

R(−2, 2, 7)

S(1, 8, 1)

#»v

#»w

#»x

#»v =

−2− 82− (−4)

7− 6

=

−1061

#»w =

−2− 112− 27− 0

=

−1307

#»x =

1− 118− 21− 0

=

−1061

Note that #»v = #»x !

Vectors in R3

Geometric Interpretation

Geometrically, a vector in R3 is represented by an arrow inxyz-space.

x

y

z

P(11, 2, 0)

Q(8,−4, 6)

R(−2, 2, 7)

S(1, 8, 1)

#»v

#»w

#»x

#»v =

−2− 82− (−4)

7− 6

=

−1061

#»w =

−2− 112− 27− 0

=

−1307

#»x =

1− 118− 21− 0

=

−1061

Note that #»v = #»x !

Vectors in R3

Geometric Interpretation

Geometrically, a vector in R3 is represented by an arrow inxyz-space.

x

y

z

P(11, 2, 0)

Q(8,−4, 6)

R(−2, 2, 7)

S(1, 8, 1)

#»v

#»w

#»x

#»v =

−2− 82− (−4)

7− 6

=

−1061

#»w =

−2− 112− 27− 0

=

−1307

#»x =

1− 118− 21− 0

=

−1061

Note that #»v = #»x !

Vectors in R3

Geometric Interpretation

Geometrically, a vector in R3 is represented by an arrow inxyz-space.

x

y

z

P(11, 2, 0)

Q(8,−4, 6)

R(−2, 2, 7)

S(1, 8, 1)

#»v

#»w

#»x

#»v =

−2− 82− (−4)

7− 6

=

−1061

#»w =

−2− 112− 27− 0

=

−1307

#»x =

1− 118− 21− 0

=

−1061

Note that #»v = #»x !

Vectors in R3

Geometric Interpretation

Geometrically, a vector in R3 is represented by an arrow inxyz-space.

x

y

z

P(11, 2, 0)

Q(8,−4, 6)

R(−2, 2, 7)

S(1, 8, 1)

#»v

#»w

#»x

#»v =

−2− 82− (−4)

7− 6

=

−1061

#»w =

−2− 112− 27− 0

=

−1307

#»x =

1− 118− 21− 0

=

−1061

Note that #»v = #»x !

Vectors in R3

Length

DefinitionThe length of #»v = 〈v1, v2, v3〉 is ‖ #»v ‖ =

√v21 + v22 + v23 .

Example

The length of #»z = 〈−7, −8, 5〉 is

‖ #»z ‖ =√

(−7)2 + (−8)2 + (5)2

=√

49 + 64 + 25

=√

138

Vectors in R3

Length

DefinitionThe length of #»v = 〈v1, v2, v3〉 is ‖ #»v ‖ =

√v21 + v22 + v23 .

Example

The length of #»z = 〈−7, −8, 5〉 is

‖ #»z ‖ =√

(−7)2 + (−8)2 + (5)2

=√

49 + 64 + 25

=√

138

Vectors in Rn

“Geometric” Interpretation

The “visible” geometry of R2 and R3 is used to define geometry inhigher dimensions.

#»a

#»

b

#»v

#»w

#»x#»y

#»z

We think of a vector #»v ∈ Rn as an “arrow” in Euclidean n-space.

Vectors in Rn

“Geometric” Interpretation

The “visible” geometry of R2 and R3 is used to define geometry inhigher dimensions.

#»a

#»

b

#»v

#»w

#»x#»y

#»z

We think of a vector #»v ∈ Rn as an “arrow” in Euclidean n-space.

Vectors in Rn

“Geometric” Interpretation

The “visible” geometry of R2 and R3 is used to define geometry inhigher dimensions.

#»a

#»

b

#»v

#»w

#»x#»y

#»z

We think of a vector #»v ∈ Rn as an “arrow” in Euclidean n-space.

Vectors in Rn

Length

The length of #»v = 〈v1, v2, . . . , vn〉 is

‖ #»v ‖ =

√v21 + v22 + · · ·+ v2n

Even if we can’t “see” a vector, we can still compute its length.

Vectors in Rn

Length

The length of #»v = 〈v1, v2, . . . , vn〉 is

‖ #»v ‖ =√v21 + v22 + · · ·+ v2n

Even if we can’t “see” a vector, we can still compute its length.

Vectors in Rn

Length

The length of #»v = 〈v1, v2, . . . , vn〉 is

‖ #»v ‖ =√v21 + v22 + · · ·+ v2n

Even if we can’t “see” a vector, we can still compute its length.

Vectors in Rn

Length

Example

The length of #»v = 〈4, 0, −2, −5, −1〉 is

‖ #»v ‖ =√

(4)2 + (0)2 + (−2)2 + (−5)2 + (−1)2

=√

16 + 0 + 4 + 25 + 1

=√

46

Example

Suppose #»a is an incidence vector of a graph on 40 nodes and 77arrows. Then #»a ∈ R40 and the length of #»a is ‖ #»a ‖ =

√2.

Vectors in Rn

Length

Example

The length of #»v = 〈4, 0, −2, −5, −1〉 is

‖ #»v ‖ =√

(4)2 + (0)2 + (−2)2 + (−5)2 + (−1)2

=√

16 + 0 + 4 + 25 + 1

=√

46

Example

Suppose #»a is an incidence vector of a graph on 40 nodes and 77arrows. Then #»a ∈

R40 and the length of #»a is ‖ #»a ‖ =√

2.

Vectors in Rn

Length

Example

The length of #»v = 〈4, 0, −2, −5, −1〉 is

‖ #»v ‖ =√

(4)2 + (0)2 + (−2)2 + (−5)2 + (−1)2

=√

16 + 0 + 4 + 25 + 1

=√

46

Example

Suppose #»a is an incidence vector of a graph on 40 nodes and 77arrows. Then #»a ∈ R40

and the length of #»a is ‖ #»a ‖ =√

2.

Vectors in Rn

Length

Example

The length of #»v = 〈4, 0, −2, −5, −1〉 is

‖ #»v ‖ =√

(4)2 + (0)2 + (−2)2 + (−5)2 + (−1)2

=√

16 + 0 + 4 + 25 + 1

=√

46

Example

Suppose #»a is an incidence vector of a graph on 40 nodes and 77arrows. Then #»a ∈ R40 and the length of #»a is ‖ #»a ‖ =

√2.

Vectors in Rn

Length

Example

The length of #»v = 〈4, 0, −2, −5, −1〉 is

‖ #»v ‖ =√

(4)2 + (0)2 + (−2)2 + (−5)2 + (−1)2

=√

16 + 0 + 4 + 25 + 1

=√

46

Example

Suppose #»a is an incidence vector of a graph on 40 nodes and 77arrows. Then #»a ∈ R40 and the length of #»a is ‖ #»a ‖ =

√2.

Scalar-Vector MultiplicationGeometric Interpretation

QuestionGiven a vector #»v ∈ Rn and a scalar c ∈ R, what “should” c · #»vlook like?

#»v2·#»v

(1/2

) ·#»v

(−1/2

) ·#»v

−1·#»v

Definition (Geometric)

The scalar-vector product of c and #»v is the vector c · #»v whosemagnitude is |c | · ‖ #»v ‖ and whose direction is either the direction of#»v if c > 0 or the opposite direction of #»v if c < 0.

Scalar-Vector MultiplicationGeometric Interpretation

QuestionGiven a vector #»v ∈ Rn and a scalar c ∈ R, what “should” c · #»vlook like?

#»v

2·#»v

(1/2

) ·#»v

(−1/2

) ·#»v

−1·#»v

Definition (Geometric)

The scalar-vector product of c and #»v is the vector c · #»v whosemagnitude is |c | · ‖ #»v ‖ and whose direction is either the direction of#»v if c > 0 or the opposite direction of #»v if c < 0.

Scalar-Vector MultiplicationGeometric Interpretation

QuestionGiven a vector #»v ∈ Rn and a scalar c ∈ R, what “should” c · #»vlook like?

#»v2·#»v

(1/2

) ·#»v

(−1/2

) ·#»v

−1·#»v

Definition (Geometric)

The scalar-vector product of c and #»v is the vector c · #»v whosemagnitude is |c | · ‖ #»v ‖ and whose direction is either the direction of#»v if c > 0 or the opposite direction of #»v if c < 0.

Scalar-Vector MultiplicationGeometric Interpretation

QuestionGiven a vector #»v ∈ Rn and a scalar c ∈ R, what “should” c · #»vlook like?

#»v2·#»v

(1/2

) ·#»v

(−1/2

) ·#»v

−1·#»v

Definition (Geometric)

The scalar-vector product of c and #»v is the vector c · #»v whosemagnitude is |c | · ‖ #»v ‖ and whose direction is either the direction of#»v if c > 0 or the opposite direction of #»v if c < 0.

Scalar-Vector MultiplicationGeometric Interpretation

QuestionGiven a vector #»v ∈ Rn and a scalar c ∈ R, what “should” c · #»vlook like?

#»v2·#»v

(1/2

) ·#»v

(−1/2

) ·#»v

−1·#»v

Definition (Geometric)

The scalar-vector product of c and #»v is the vector c · #»v whosemagnitude is |c | · ‖ #»v ‖ and whose direction is either the direction of#»v if c > 0 or the opposite direction of #»v if c < 0.

Scalar-Vector MultiplicationGeometric Interpretation

QuestionGiven a vector #»v ∈ Rn and a scalar c ∈ R, what “should” c · #»vlook like?

#»v2·#»v

(1/2

) ·#»v

(−1/2

) ·#»v

−1·#»v

Definition (Geometric)

The scalar-vector product of c and #»v is the vector c · #»v whosemagnitude is |c | · ‖ #»v ‖ and whose direction is either the direction of#»v if c > 0 or the opposite direction of #»v if c < 0.

Scalar-Vector MultiplicationGeometric Interpretation

QuestionGiven a vector #»v ∈ Rn and a scalar c ∈ R, what “should” c · #»vlook like?

#»v2·#»v

(1/2

) ·#»v

(−1/2

) ·#»v

−1·#»v

Definition (Geometric)

The scalar-vector product of c and #»v is the vector c · #»v whosemagnitude is |c | · ‖ #»v ‖ and whose direction is either the direction of#»v if c > 0 or the opposite direction of #»v if c < 0.

Scalar-Vector MultiplicationAlgebraic Comparison

This “geometric” interpretation coincides with our “algebraic”definition.

‖c · #»v ‖ = ‖〈c · v1, c · v2, . . . , c · vn〉‖

=√

(c · v1)2 + (c · v2)2 + · · ·+ (c · vn)2

=√

c2 · v21 + c2 · v22 + · · ·+ c2 · v2n

=√

c2 · (v21 + v22 + · · ·+ v2n )

=√c2 ·

√v21 + v22 + · · ·+ v2n

= |c | · ‖ #»v ‖

Scalar-Vector MultiplicationAlgebraic Comparison

This “geometric” interpretation coincides with our “algebraic”definition.

‖c · #»v ‖ = ‖〈c · v1, c · v2, . . . , c · vn〉‖

=√

(c · v1)2 + (c · v2)2 + · · ·+ (c · vn)2

=√

c2 · v21 + c2 · v22 + · · ·+ c2 · v2n

=√

c2 · (v21 + v22 + · · ·+ v2n )

=√c2 ·

√v21 + v22 + · · ·+ v2n

= |c | · ‖ #»v ‖

Scalar-Vector MultiplicationAlgebraic Comparison

This “geometric” interpretation coincides with our “algebraic”definition.

‖c · #»v ‖ = ‖〈c · v1, c · v2, . . . , c · vn〉‖

=√

(c · v1)2 + (c · v2)2 + · · ·+ (c · vn)2

=√

c2 · v21 + c2 · v22 + · · ·+ c2 · v2n

=√

c2 · (v21 + v22 + · · ·+ v2n )

=√c2 ·

√v21 + v22 + · · ·+ v2n

= |c | · ‖ #»v ‖

Scalar-Vector MultiplicationAlgebraic Comparison

This “geometric” interpretation coincides with our “algebraic”definition.

‖c · #»v ‖ = ‖〈c · v1, c · v2, . . . , c · vn〉‖

=√

(c · v1)2 + (c · v2)2 + · · ·+ (c · vn)2

=√

c2 · v21 + c2 · v22 + · · ·+ c2 · v2n

=√

c2 · (v21 + v22 + · · ·+ v2n )

=√c2 ·

√v21 + v22 + · · ·+ v2n

= |c | · ‖ #»v ‖

Scalar-Vector MultiplicationAlgebraic Comparison

This “geometric” interpretation coincides with our “algebraic”definition.

‖c · #»v ‖ = ‖〈c · v1, c · v2, . . . , c · vn〉‖

=√

(c · v1)2 + (c · v2)2 + · · ·+ (c · vn)2

=√

c2 · v21 + c2 · v22 + · · ·+ c2 · v2n

=√

c2 · (v21 + v22 + · · ·+ v2n )

=√c2 ·

√v21 + v22 + · · ·+ v2n

= |c | · ‖ #»v ‖

Scalar-Vector MultiplicationAlgebraic Comparison

This “geometric” interpretation coincides with our “algebraic”definition.

‖c · #»v ‖ = ‖〈c · v1, c · v2, . . . , c · vn〉‖

=√

(c · v1)2 + (c · v2)2 + · · ·+ (c · vn)2

=√

c2 · v21 + c2 · v22 + · · ·+ c2 · v2n

=√

c2 · (v21 + v22 + · · ·+ v2n )

=√c2 ·

√v21 + v22 + · · ·+ v2n

= |c | · ‖ #»v ‖

Scalar-Vector MultiplicationUnit Vectors

DefinitionA unit vector is a vector with length one.

Example

Let #»u = 〈1/2, −1/2, −1/2, 1/2〉 . Then

‖ #»u ‖

= ‖(1/2) · 〈1, −1, −1, 1〉 ‖= |1/2| · ‖〈1, −1, −1, 1〉 ‖

= (1/2) ·√

(1)2 + (−1)2 + (−1)2 + (1)2

= (1/2) ·√

1 + 1 + 1 + 1

= (1/2) ·√

4

= (1/2) · 2= 1

Scalar-Vector MultiplicationUnit Vectors

DefinitionA unit vector is a vector with length one.

Example

Let #»u = 〈1/2, −1/2, −1/2, 1/2〉 . Then

‖ #»u ‖ = ‖(1/2) · 〈1, −1, −1, 1〉 ‖

= |1/2| · ‖〈1, −1, −1, 1〉 ‖

= (1/2) ·√

(1)2 + (−1)2 + (−1)2 + (1)2

= (1/2) ·√

1 + 1 + 1 + 1

= (1/2) ·√

4

= (1/2) · 2= 1

Scalar-Vector MultiplicationUnit Vectors

DefinitionA unit vector is a vector with length one.

Example

Let #»u = 〈1/2, −1/2, −1/2, 1/2〉 . Then

‖ #»u ‖ = ‖(1/2) · 〈1, −1, −1, 1〉 ‖= |1/2| · ‖〈1, −1, −1, 1〉 ‖

= (1/2) ·√

(1)2 + (−1)2 + (−1)2 + (1)2

= (1/2) ·√

1 + 1 + 1 + 1

= (1/2) ·√

4

= (1/2) · 2= 1

Scalar-Vector MultiplicationUnit Vectors

DefinitionA unit vector is a vector with length one.

Example

Let #»u = 〈1/2, −1/2, −1/2, 1/2〉 . Then

‖ #»u ‖ = ‖(1/2) · 〈1, −1, −1, 1〉 ‖= |1/2| · ‖〈1, −1, −1, 1〉 ‖

= (1/2) ·√

(1)2 + (−1)2 + (−1)2 + (1)2

= (1/2) ·√

1 + 1 + 1 + 1

= (1/2) ·√

4

= (1/2) · 2= 1

Scalar-Vector MultiplicationUnit Vectors

DefinitionA unit vector is a vector with length one.

Example

Let #»u = 〈1/2, −1/2, −1/2, 1/2〉 . Then

‖ #»u ‖ = ‖(1/2) · 〈1, −1, −1, 1〉 ‖= |1/2| · ‖〈1, −1, −1, 1〉 ‖

= (1/2) ·√

(1)2 + (−1)2 + (−1)2 + (1)2

= (1/2) ·√

1 + 1 + 1 + 1

= (1/2) ·√

4

= (1/2) · 2= 1

Scalar-Vector MultiplicationUnit Vectors

DefinitionA unit vector is a vector with length one.

Example

Let #»u = 〈1/2, −1/2, −1/2, 1/2〉 . Then

‖ #»u ‖ = ‖(1/2) · 〈1, −1, −1, 1〉 ‖= |1/2| · ‖〈1, −1, −1, 1〉 ‖

= (1/2) ·√

(1)2 + (−1)2 + (−1)2 + (1)2

= (1/2) ·√

1 + 1 + 1 + 1

= (1/2) ·√

4

= (1/2) · 2= 1

Scalar-Vector MultiplicationUnit Vectors

DefinitionA unit vector is a vector with length one.

Example

Let #»u = 〈1/2, −1/2, −1/2, 1/2〉 . Then

‖ #»u ‖ = ‖(1/2) · 〈1, −1, −1, 1〉 ‖= |1/2| · ‖〈1, −1, −1, 1〉 ‖

= (1/2) ·√

(1)2 + (−1)2 + (−1)2 + (1)2

= (1/2) ·√

1 + 1 + 1 + 1

= (1/2) ·√

4

= (1/2) · 2

= 1

Scalar-Vector MultiplicationUnit Vectors

DefinitionA unit vector is a vector with length one.

Example

Let #»u = 〈1/2, −1/2, −1/2, 1/2〉 . Then

‖ #»u ‖ = ‖(1/2) · 〈1, −1, −1, 1〉 ‖= |1/2| · ‖〈1, −1, −1, 1〉 ‖

= (1/2) ·√

(1)2 + (−1)2 + (−1)2 + (1)2

= (1/2) ·√

1 + 1 + 1 + 1

= (1/2) ·√

4

= (1/2) · 2= 1

Scalar-Vector MultiplicationUnit Vectors

Example

Consider the vectors

#»e 1 = 〈1, 0, 0〉 #»e 2 = 〈0, 1, 0〉 #»e 3 = 〈0, 0, 1〉

Each #»e 1, #»e 2, and #»e 3 is a unit vector.

DefinitionThese vectors form the standard basis of R3.

Important

Every vector #»v = 〈v1, v2, v3〉 satisfies

#»v = v1 · #»e 1 + v2 · #»e 2 + v3 · #»e 3

Every vector is a linear combination of the standard basis!

Scalar-Vector MultiplicationUnit Vectors

Example

Consider the vectors

#»e 1 = 〈1, 0, 0〉 #»e 2 = 〈0, 1, 0〉 #»e 3 = 〈0, 0, 1〉

Each #»e 1, #»e 2, and #»e 3 is a unit vector.

DefinitionThese vectors form the standard basis of R3.

Important

Every vector #»v = 〈v1, v2, v3〉 satisfies

#»v = v1 · #»e 1 + v2 · #»e 2 + v3 · #»e 3

Every vector is a linear combination of the standard basis!

Scalar-Vector MultiplicationUnit Vectors

Example

Consider the vectors

#»e 1 = 〈1, 0, 0〉 #»e 2 = 〈0, 1, 0〉 #»e 3 = 〈0, 0, 1〉

Each #»e 1, #»e 2, and #»e 3 is a unit vector.

DefinitionThese vectors form the standard basis of R3.

Important

Every vector #»v = 〈v1, v2, v3〉 satisfies

#»v = v1 · #»e 1 + v2 · #»e 2 + v3 · #»e 3

Every vector is a linear combination of the standard basis!

Scalar-Vector MultiplicationUnit Vectors

Example

Consider the vectors

#»e 1 = 〈1, 0, 0〉 #»e 2 = 〈0, 1, 0〉 #»e 3 = 〈0, 0, 1〉

Each #»e 1, #»e 2, and #»e 3 is a unit vector.

DefinitionThese vectors form the standard basis of R3.

Important

Every vector #»v = 〈v1, v2, v3〉 satisfies

#»v = v1 · #»e 1 + v2 · #»e 2 + v3 · #»e 3

Every vector is a

linear combination of the standard basis!

Scalar-Vector MultiplicationUnit Vectors

Example

Consider the vectors

#»e 1 = 〈1, 0, 0〉 #»e 2 = 〈0, 1, 0〉 #»e 3 = 〈0, 0, 1〉

Each #»e 1, #»e 2, and #»e 3 is a unit vector.

DefinitionThese vectors form the standard basis of R3.

Important

Every vector #»v = 〈v1, v2, v3〉 satisfies

#»v = v1 · #»e 1 + v2 · #»e 2 + v3 · #»e 3

Every vector is a linear combination of the standard basis!

Scalar-Vector MultiplicationNormalization

DefinitionThe normalization of #»v is v̂ = (1/‖ #»v ‖) · #»v .

#»v

v̂

Note that

‖v̂‖ =

∥∥∥∥ 1

‖ #»v ‖· #»v

∥∥∥∥ =

∣∣∣∣ 1

‖ #»v ‖

∣∣∣∣ · ‖ #»v ‖ =1

‖ #»v ‖· ‖ #»v ‖ = 1

The normalization v̂ of #»v is always a unit vector!

Scalar-Vector MultiplicationNormalization

DefinitionThe normalization of #»v is v̂ = (1/‖ #»v ‖) · #»v .

#»v

v̂

Note that

‖v̂‖ =

∥∥∥∥ 1

‖ #»v ‖· #»v

∥∥∥∥ =

∣∣∣∣ 1

‖ #»v ‖

∣∣∣∣ · ‖ #»v ‖ =1

‖ #»v ‖· ‖ #»v ‖ = 1

The normalization v̂ of #»v is always a unit vector!

Scalar-Vector MultiplicationNormalization

DefinitionThe normalization of #»v is v̂ = (1/‖ #»v ‖) · #»v .

#»v

v̂

Note that

‖v̂‖ =

∥∥∥∥ 1

‖ #»v ‖· #»v

∥∥∥∥ =

∣∣∣∣ 1

‖ #»v ‖

∣∣∣∣ · ‖ #»v ‖ =1

‖ #»v ‖· ‖ #»v ‖ = 1

The normalization v̂ of #»v is always a unit vector!

Scalar-Vector MultiplicationNormalization

DefinitionThe normalization of #»v is v̂ = (1/‖ #»v ‖) · #»v .

#»v

v̂

Note that

‖v̂‖ =

∥∥∥∥ 1

‖ #»v ‖· #»v

∥∥∥∥ =

∣∣∣∣ 1

‖ #»v ‖

∣∣∣∣ · ‖ #»v ‖ =1

‖ #»v ‖· ‖ #»v ‖ = 1

The normalization v̂ of #»v is always a unit vector!

Scalar-Vector MultiplicationNormalization

DefinitionThe normalization of #»v is v̂ = (1/‖ #»v ‖) · #»v .

#»v

v̂

Note that

‖v̂‖ =

∥∥∥∥ 1

‖ #»v ‖· #»v

∥∥∥∥ =

∣∣∣∣ 1

‖ #»v ‖

∣∣∣∣ · ‖ #»v ‖ =

1

‖ #»v ‖· ‖ #»v ‖ = 1

The normalization v̂ of #»v is always a unit vector!

Scalar-Vector MultiplicationNormalization

DefinitionThe normalization of #»v is v̂ = (1/‖ #»v ‖) · #»v .

#»v

v̂

Note that

‖v̂‖ =

∥∥∥∥ 1

‖ #»v ‖· #»v

∥∥∥∥ =

∣∣∣∣ 1

‖ #»v ‖

∣∣∣∣ · ‖ #»v ‖ =1

‖ #»v ‖· ‖ #»v ‖ =

1

The normalization v̂ of #»v is always a unit vector!

Scalar-Vector MultiplicationNormalization

DefinitionThe normalization of #»v is v̂ = (1/‖ #»v ‖) · #»v .

#»v

v̂

Note that

‖v̂‖ =

∥∥∥∥ 1

‖ #»v ‖· #»v

∥∥∥∥ =

∣∣∣∣ 1

‖ #»v ‖

∣∣∣∣ · ‖ #»v ‖ =1

‖ #»v ‖· ‖ #»v ‖ = 1

The normalization v̂ of #»v is always a unit vector!

Scalar-Vector MultiplicationNormalization

DefinitionThe normalization of #»v is v̂ = (1/‖ #»v ‖) · #»v .

#»v

v̂

Note that

‖v̂‖ =

∥∥∥∥ 1

‖ #»v ‖· #»v

∥∥∥∥ =

∣∣∣∣ 1

‖ #»v ‖

∣∣∣∣ · ‖ #»v ‖ =1

‖ #»v ‖· ‖ #»v ‖ = 1

The normalization v̂ of #»v is always a unit vector!

Vector AdditionGeometric Interpretation

To interpret #»v + #»w geometrically, start by forming a parallelogram.

The sum #»v + #»w is the diagonal of the parallelogram, obtained byfirst transversing #»v and then transversing #»w .

#»v

#»w

#»v

#»w#»v

+#»w

Vector AdditionGeometric Interpretation

To interpret #»v + #»w geometrically, start by forming a parallelogram.

The sum #»v + #»w is the diagonal of the parallelogram, obtained byfirst transversing #»v and then transversing #»w .

#»v

#»w

#»v

#»w#»v

+#»w

Vector AdditionGeometric Interpretation

To interpret #»v + #»w geometrically, start by forming a parallelogram.

The sum #»v + #»w is the diagonal of the parallelogram, obtained byfirst transversing #»v and then transversing #»w .

#»v

#»w

#»v

#»w#»v

+#»w

Vector AdditionGeometric Interpretation

To interpret #»v + #»w geometrically, start by forming a parallelogram.

The sum #»v + #»w is the diagonal of the parallelogram, obtained byfirst transversing #»v and then transversing #»w .

#»v

#»w

#»v

#»w

#»v+

#»w

Vector AdditionGeometric Interpretation

To interpret #»v + #»w geometrically, start by forming a parallelogram.The sum #»v + #»w is the diagonal of the parallelogram, obtained byfirst transversing #»v and then transversing #»w .

#»v

#»w

#»v

#»w

#»v+

#»w

Vector AdditionGeometric Interpretation

To interpret #»v + #»w geometrically, start by forming a parallelogram.The sum #»v + #»w is the diagonal of the parallelogram, obtained byfirst transversing #»v and then transversing #»w .

#»v

#»w

#»v

#»w#»v

+#»w

Vector AdditionGeometric Interpretation

This geometric interpretation allows us to construct vectordiagrams.

#»v#»w#»x

#»y =

#»v − #»x

#»z =

#»w − #»v

Vector AdditionGeometric Interpretation

This geometric interpretation allows us to construct vectordiagrams.

#»v#»w#»x

#»y = #»v − #»x

#»z =

#»w − #»v

Vector AdditionGeometric Interpretation

This geometric interpretation allows us to construct vectordiagrams.

#»v#»w#»x

#»y = #»v − #»x

#»z = #»w − #»v

Vector AdditionProperties

Properties of Vector Arithmetic

Vector arithmetic obeys the following laws.

additive commutativity #»v + #»w = #»w + #»v

additive associativity ( #»v + #»w ) + #»x = #»v + ( #»w + #»x )

multiplicitive associativity (c · d) · #»v = c · (d · #»v )

scalar distribution c · ( #»v + #»w ) = c · #»v + c · #»w

vector distribution (c + d) · #»v = c · #»v + d · #»v

additive identity the zero vector#»

O = 〈0, 0, . . . , 0〉satisfies the equation

#»

O + #»v = #»v

NoteThe zero vector

#»

O is the only vector whose length is zero ‖ #»

O‖ = 0.

Vector AdditionProperties

Properties of Vector Arithmetic

Vector arithmetic obeys the following laws.

additive commutativity #»v + #»w = #»w + #»v

additive associativity ( #»v + #»w ) + #»x = #»v + ( #»w + #»x )

multiplicitive associativity (c · d) · #»v = c · (d · #»v )

scalar distribution c · ( #»v + #»w ) = c · #»v + c · #»w

vector distribution (c + d) · #»v = c · #»v + d · #»v

additive identity the zero vector#»

O = 〈0, 0, . . . , 0〉satisfies the equation

#»

O + #»v = #»v

NoteThe zero vector

#»

O is the only vector whose length is zero ‖ #»

O‖ = 0.

Vector AdditionProperties

Properties of Vector Arithmetic

Vector arithmetic obeys the following laws.

additive commutativity #»v + #»w = #»w + #»v

additive associativity ( #»v + #»w ) + #»x = #»v + ( #»w + #»x )

multiplicitive associativity (c · d) · #»v = c · (d · #»v )

scalar distribution c · ( #»v + #»w ) = c · #»v + c · #»w

vector distribution (c + d) · #»v = c · #»v + d · #»v

additive identity the zero vector#»

O = 〈0, 0, . . . , 0〉satisfies the equation

#»

O + #»v = #»v

NoteThe zero vector

#»

O is the only vector whose length is zero ‖ #»

O‖ = 0.

Vector AdditionProperties

Properties of Vector Arithmetic

Vector arithmetic obeys the following laws.

additive commutativity #»v + #»w = #»w + #»v

additive associativity ( #»v + #»w ) + #»x = #»v + ( #»w + #»x )

multiplicitive associativity (c · d) · #»v = c · (d · #»v )

scalar distribution c · ( #»v + #»w ) = c · #»v + c · #»w

vector distribution (c + d) · #»v = c · #»v + d · #»v

additive identity the zero vector#»

O = 〈0, 0, . . . , 0〉satisfies the equation

#»

O + #»v = #»v

NoteThe zero vector

#»

O is the only vector whose length is zero ‖ #»

O‖ = 0.

Vector AdditionProperties

Properties of Vector Arithmetic

Vector arithmetic obeys the following laws.

additive commutativity #»v + #»w = #»w + #»v

additive associativity ( #»v + #»w ) + #»x = #»v + ( #»w + #»x )

multiplicitive associativity (c · d) · #»v = c · (d · #»v )

scalar distribution c · ( #»v + #»w ) = c · #»v + c · #»w

vector distribution (c + d) · #»v = c · #»v + d · #»v

additive identity the zero vector#»

O = 〈0, 0, . . . , 0〉satisfies the equation

#»

O + #»v = #»v

NoteThe zero vector

#»

O is the only vector whose length is zero ‖ #»

O‖ = 0.

Vector AdditionProperties

Properties of Vector Arithmetic

Vector arithmetic obeys the following laws.

additive commutativity #»v + #»w = #»w + #»v

additive associativity ( #»v + #»w ) + #»x = #»v + ( #»w + #»x )

multiplicitive associativity (c · d) · #»v = c · (d · #»v )

scalar distribution c · ( #»v + #»w ) = c · #»v + c · #»w

vector distribution (c + d) · #»v = c · #»v + d · #»v

additive identity the zero vector#»

O = 〈0, 0, . . . , 0〉satisfies the equation

#»

O + #»v = #»v

NoteThe zero vector

#»

O is the only vector whose length is zero ‖ #»

O‖ = 0.

Vector AdditionProperties

Properties of Vector Arithmetic

Vector arithmetic obeys the following laws.

additive commutativity #»v + #»w = #»w + #»v

additive associativity ( #»v + #»w ) + #»x = #»v + ( #»w + #»x )

multiplicitive associativity (c · d) · #»v = c · (d · #»v )

scalar distribution c · ( #»v + #»w ) = c · #»v + c · #»w

vector distribution (c + d) · #»v = c · #»v + d · #»v

additive identity the zero vector#»

O = 〈0, 0, . . . , 0〉satisfies the equation

#»

O + #»v = #»v

NoteThe zero vector

#»

O is the only vector whose length is zero ‖ #»

O‖ = 0.

The Dot ProductDefinition

QuestionWhat does it mean to multiply two vectors?

AnswerThere is no correct answer. There are several useful ways tomultiply vectors.

The Dot ProductDefinition

QuestionWhat does it mean to multiply two vectors?

AnswerThere is no correct answer. There are several useful ways tomultiply vectors.

The Dot ProductDefinition

DefinitionThe dot product of #»v = 〈v1, v2, . . . , vn〉 and #»w = 〈w1,w2, . . . ,wn〉is defined as

#»v · #»w = v1 · w1 + v2 · w2 + · · ·+ vn · wn

NoteThe dot product of two vectors is a scalar, not a vector.

NoteThe dot product #»v · #»w is only defined if #»v and #»w have the samenumber of coordinates.

The Dot ProductDefinition

DefinitionThe dot product of #»v = 〈v1, v2, . . . , vn〉 and #»w = 〈w1,w2, . . . ,wn〉is defined as

#»v · #»w = v1 · w1 + v2 · w2 + · · ·+ vn · wn

NoteThe dot product of two vectors is a scalar, not a vector.

NoteThe dot product #»v · #»w is only defined if #»v and #»w have the samenumber of coordinates.

The Dot ProductDefinition

DefinitionThe dot product of #»v = 〈v1, v2, . . . , vn〉 and #»w = 〈w1,w2, . . . ,wn〉is defined as

#»v · #»w = v1 · w1 + v2 · w2 + · · ·+ vn · wn

NoteThe dot product of two vectors is a scalar, not a vector.

NoteThe dot product #»v · #»w is only defined if #»v and #»w have the samenumber of coordinates.

The Dot ProductDefinition

Example

Let #»v = 〈−1, −6, 1〉 and #»w = 〈−2, −3, −4〉 . Then

#»v · #»w = 〈−1, −6, 1〉 · 〈−2, −3, −4〉= (−1)(−2) + (−6)(−3) + (1)(−4)

= (2) + (18) + (−4)

= 16

Example

The dot product of

1−2

0

and

−1−2−1

0

is not defined!

The Dot ProductDefinition

Example

Let #»v = 〈−1, −6, 1〉 and #»w = 〈−2, −3, −4〉 . Then

#»v · #»w =

〈−1, −6, 1〉 · 〈−2, −3, −4〉= (−1)(−2) + (−6)(−3) + (1)(−4)

= (2) + (18) + (−4)

= 16

Example

The dot product of

1−2

0

and

−1−2−1

0

is not defined!

The Dot ProductDefinition

Example

Let #»v = 〈−1, −6, 1〉 and #»w = 〈−2, −3, −4〉 . Then

#»v · #»w = 〈−1, −6, 1〉 · 〈−2, −3, −4〉=

(−1)(−2) + (−6)(−3) + (1)(−4)

= (2) + (18) + (−4)

= 16

Example

The dot product of

1−2

0

and

−1−2−1

0

is not defined!

The Dot ProductDefinition

Example

Let #»v = 〈−1, −6, 1〉 and #»w = 〈−2, −3, −4〉 . Then

#»v · #»w = 〈−1, −6, 1〉 · 〈−2, −3, −4〉= (−1)(−2) + (−6)(−3) + (1)(−4)

=

(2) + (18) + (−4)

= 16

Example

The dot product of

1−2

0

and

−1−2−1

0

is not defined!

The Dot ProductDefinition

Example

Let #»v = 〈−1, −6, 1〉 and #»w = 〈−2, −3, −4〉 . Then

#»v · #»w = 〈−1, −6, 1〉 · 〈−2, −3, −4〉= (−1)(−2) + (−6)(−3) + (1)(−4)

= (2) + (18) + (−4)

=

16

Example

The dot product of

1−2

0

and

−1−2−1

0

is not defined!

The Dot ProductDefinition

Example

Let #»v = 〈−1, −6, 1〉 and #»w = 〈−2, −3, −4〉 . Then

#»v · #»w = 〈−1, −6, 1〉 · 〈−2, −3, −4〉= (−1)(−2) + (−6)(−3) + (1)(−4)

= (2) + (18) + (−4)

= 16

Example

The dot product of

1−2

0

and

−1−2−1

0

is not defined!

The Dot ProductDefinition

Example

Let #»v = 〈−1, −6, 1〉 and #»w = 〈−2, −3, −4〉 . Then

#»v · #»w = 〈−1, −6, 1〉 · 〈−2, −3, −4〉= (−1)(−2) + (−6)(−3) + (1)(−4)

= (2) + (18) + (−4)

= 16

Example

The dot product of

1−2

0

and

−1−2−1

0

is not defined!

The Dot ProductProperties

Algebraic Properties of the Dot Product

The dot product obeys the following laws.

commutative #»v · #»w = #»w · #»v

distributive #»v · ( #»w + #»x ) = #»v · #»w + #»v · #»x

associative c · ( #»v · #»w ) = (c · #»v ) · #»w

The Dot ProductProperties

Algebraic Properties of the Dot Product

The dot product obeys the following laws.

commutative #»v · #»w = #»w · #»v

distributive #»v · ( #»w + #»x ) = #»v · #»w + #»v · #»x

associative c · ( #»v · #»w ) = (c · #»v ) · #»w

The Dot ProductProperties

Algebraic Properties of the Dot Product

The dot product obeys the following laws.

commutative #»v · #»w = #»w · #»v

distributive #»v · ( #»w + #»x ) = #»v · #»w + #»v · #»x

associative c · ( #»v · #»w ) = (c · #»v ) · #»w

The Dot ProductLengths

ObservationFor #»v = 〈v1, v2, . . . , vn〉, we have

#»v · #»v = v1 · v1 + v2 · v2 + · · ·+ vn · vn= v21 + v22 + · · ·+ v2n

= ‖ #»v ‖2

The dot product can be used to compute lengths!

The Dot ProductLengths

ObservationFor #»v = 〈v1, v2, . . . , vn〉, we have

#»v · #»v =

v1 · v1 + v2 · v2 + · · ·+ vn · vn= v21 + v22 + · · ·+ v2n

= ‖ #»v ‖2

The dot product can be used to compute lengths!

The Dot ProductLengths

ObservationFor #»v = 〈v1, v2, . . . , vn〉, we have

#»v · #»v = v1 · v1 + v2 · v2 + · · ·+ vn · vn=

v21 + v22 + · · ·+ v2n

= ‖ #»v ‖2

The dot product can be used to compute lengths!

The Dot ProductLengths

ObservationFor #»v = 〈v1, v2, . . . , vn〉, we have

#»v · #»v = v1 · v1 + v2 · v2 + · · ·+ vn · vn= v21 + v22 + · · ·+ v2n

=

‖ #»v ‖2

The dot product can be used to compute lengths!

The Dot ProductLengths

ObservationFor #»v = 〈v1, v2, . . . , vn〉, we have

#»v · #»v = v1 · v1 + v2 · v2 + · · ·+ vn · vn= v21 + v22 + · · ·+ v2n

= ‖ #»v ‖2

The dot product can be used to compute lengths!

The Dot ProductLengths

ObservationFor #»v = 〈v1, v2, . . . , vn〉, we have

#»v · #»v = v1 · v1 + v2 · v2 + · · ·+ vn · vn= v21 + v22 + · · ·+ v2n

= ‖ #»v ‖2

The dot product can be used to compute lengths!

The Dot ProductAngles

QuestionHow can we measure the angle θbetween two vectors #»v and #»w?

#»v

#»w

θ

#»v − #»w

AnswerForm a triangle. Measure ‖ #»v − #»w‖2 in two ways.

law of cosines ‖ #»v − #»w‖2 = ‖ #»v ‖2 + ‖ #»w‖2 − 2 ‖ #»v ‖ · ‖ #»w‖ cos θ

dot product ‖ #»v − #»w‖2 = ‖ #»v ‖2 + ‖ #»w‖2 − 2 #»v · #»w

Theorem#»v · #»w = ‖ #»v ‖ · ‖ #»w‖ cos θ

The Dot ProductAngles

QuestionHow can we measure the angle θbetween two vectors #»v and #»w?

#»v

#»w

θ

#»v − #»w

AnswerForm a triangle.

Measure ‖ #»v − #»w‖2 in two ways.

law of cosines ‖ #»v − #»w‖2 = ‖ #»v ‖2 + ‖ #»w‖2 − 2 ‖ #»v ‖ · ‖ #»w‖ cos θ

dot product ‖ #»v − #»w‖2 = ‖ #»v ‖2 + ‖ #»w‖2 − 2 #»v · #»w

Theorem#»v · #»w = ‖ #»v ‖ · ‖ #»w‖ cos θ

The Dot ProductAngles

QuestionHow can we measure the angle θbetween two vectors #»v and #»w?

#»v

#»w

θ

#»v − #»w

AnswerForm a triangle.

Measure ‖ #»v − #»w‖2 in two ways.

law of cosines ‖ #»v − #»w‖2 = ‖ #»v ‖2 + ‖ #»w‖2 − 2 ‖ #»v ‖ · ‖ #»w‖ cos θ

dot product ‖ #»v − #»w‖2 = ‖ #»v ‖2 + ‖ #»w‖2 − 2 #»v · #»w

Theorem#»v · #»w = ‖ #»v ‖ · ‖ #»w‖ cos θ

The Dot ProductAngles

QuestionHow can we measure the angle θbetween two vectors #»v and #»w?

#»v

#»w

θ

#»v − #»w

AnswerForm a triangle. Measure ‖ #»v − #»w‖2 in two ways.

law of cosines ‖ #»v − #»w‖2 = ‖ #»v ‖2 + ‖ #»w‖2 − 2 ‖ #»v ‖ · ‖ #»w‖ cos θ

dot product ‖ #»v − #»w‖2 = ‖ #»v ‖2 + ‖ #»w‖2 − 2 #»v · #»w

Theorem#»v · #»w = ‖ #»v ‖ · ‖ #»w‖ cos θ

The Dot ProductAngles

QuestionHow can we measure the angle θbetween two vectors #»v and #»w?

#»v

#»w

θ

#»v − #»w

AnswerForm a triangle. Measure ‖ #»v − #»w‖2 in two ways.

law of cosines ‖ #»v − #»w‖2 = ‖ #»v ‖2 + ‖ #»w‖2 − 2 ‖ #»v ‖ · ‖ #»w‖ cos θ

dot product ‖ #»v − #»w‖2 = ‖ #»v ‖2 + ‖ #»w‖2 − 2 #»v · #»w

Theorem#»v · #»w = ‖ #»v ‖ · ‖ #»w‖ cos θ

The Dot ProductAngles

QuestionHow can we measure the angle θbetween two vectors #»v and #»w?

#»v

#»w

θ

#»v − #»w

AnswerForm a triangle. Measure ‖ #»v − #»w‖2 in two ways.

law of cosines ‖ #»v − #»w‖2 = ‖ #»v ‖2 + ‖ #»w‖2 − 2 ‖ #»v ‖ · ‖ #»w‖ cos θ

dot product ‖ #»v − #»w‖2 = ‖ #»v ‖2 + ‖ #»w‖2 − 2 #»v · #»w

Theorem#»v · #»w = ‖ #»v ‖ · ‖ #»w‖ cos θ

The Dot ProductAngles

QuestionHow can we measure the angle θbetween two vectors #»v and #»w?

#»v

#»w

θ

#»v − #»w

AnswerForm a triangle. Measure ‖ #»v − #»w‖2 in two ways.

law of cosines ‖ #»v − #»w‖2 = ‖ #»v ‖2 + ‖ #»w‖2 − 2 ‖ #»v ‖ · ‖ #»w‖ cos θ

dot product ‖ #»v − #»w‖2 = ‖ #»v ‖2 + ‖ #»w‖2 − 2 #»v · #»w

Theorem#»v · #»w = ‖ #»v ‖ · ‖ #»w‖ cos θ

The Dot ProductAngles

Theorem#»v · #»w = ‖ #»v ‖ · ‖ #»w‖ cos θ

Corollary (The Cauchy-Schwarz Inequality)

| #»v · #»w | ≤ ‖ #»v ‖ · ‖ #»w‖

Example

Let θ be the angle between #»v = 〈1, 2, 3〉 and #»w = 〈1, 1, 1〉.Compute cos θ.

cos θ =#»v · #»w

‖ #»v ‖ · ‖ #»w‖=

(1)(1) + (2)(1) + (3)(1)√12 + 22 + 32 ·

√12 + 12 + 12

=6√

14 ·√

3

The Dot ProductAngles

Theorem#»v · #»w = ‖ #»v ‖ · ‖ #»w‖ cos θ

Corollary (The Cauchy-Schwarz Inequality)

| #»v · #»w | ≤ ‖ #»v ‖ · ‖ #»w‖

Example

Let θ be the angle between #»v = 〈1, 2, 3〉 and #»w = 〈1, 1, 1〉.Compute cos θ.

cos θ =#»v · #»w

‖ #»v ‖ · ‖ #»w‖=

(1)(1) + (2)(1) + (3)(1)√12 + 22 + 32 ·

√12 + 12 + 12

=6√

14 ·√

3

The Dot ProductAngles

Theorem#»v · #»w = ‖ #»v ‖ · ‖ #»w‖ cos θ

Corollary (The Cauchy-Schwarz Inequality)

| #»v · #»w | ≤ ‖ #»v ‖ · ‖ #»w‖

Example

Let θ be the angle between #»v = 〈1, 2, 3〉 and #»w = 〈1, 1, 1〉.Compute cos θ.

cos θ =#»v · #»w

‖ #»v ‖ · ‖ #»w‖=

(1)(1) + (2)(1) + (3)(1)√12 + 22 + 32 ·

√12 + 12 + 12

=6√

14 ·√

3

The Dot ProductAngles

Theorem#»v · #»w = ‖ #»v ‖ · ‖ #»w‖ cos θ

Corollary (The Cauchy-Schwarz Inequality)

| #»v · #»w | ≤ ‖ #»v ‖ · ‖ #»w‖

Example

Let θ be the angle between #»v = 〈1, 2, 3〉 and #»w = 〈1, 1, 1〉.Compute cos θ.

cos θ =#»v · #»w

‖ #»v ‖ · ‖ #»w‖=

(1)(1) + (2)(1) + (3)(1)√12 + 22 + 32 ·

√12 + 12 + 12

=6√

14 ·√

3

The Dot ProductAngles

TheoremLet θ be the angle between two vectors #»v 6= #»

O and #»w 6= #»

O . Then

#»v · #»w > 0 means θ is

acute (0 ≤ θ < π/2)

#»v · #»w < 0 means θ is

obtuse (π/2 < θ ≤ π)

#»v · #»w = 0 means θ is

right (θ = π/2)

Definition#»v and #»w are orthogonal if #»v · #»w = 0

Example

Are #»v = 〈1,√

2, 1, 0〉 and #»w = 〈1,−√

2, 1, 1〉 orthogonal? Yes,since

#»v · #»w = (1)(1) + (√

2)(−√

2) + (1)(1) + (0)(1) = 1− 2 + 1 + 0 = 0

The Dot ProductAngles

TheoremLet θ be the angle between two vectors #»v 6= #»

O and #»w 6= #»

O . Then#»v · #»w > 0 means θ is

acute (0 ≤ θ < π/2)#»v · #»w < 0 means θ is

obtuse (π/2 < θ ≤ π)

#»v · #»w = 0 means θ is

right (θ = π/2)

Definition#»v and #»w are orthogonal if #»v · #»w = 0

Example

Are #»v = 〈1,√

2, 1, 0〉 and #»w = 〈1,−√

2, 1, 1〉 orthogonal? Yes,since

#»v · #»w = (1)(1) + (√

2)(−√

2) + (1)(1) + (0)(1) = 1− 2 + 1 + 0 = 0

The Dot ProductAngles

TheoremLet θ be the angle between two vectors #»v 6= #»

O and #»w 6= #»

O . Then#»v · #»w > 0 means θ is acute (0 ≤ θ < π/2)

#»v · #»w < 0 means θ is

obtuse (π/2 < θ ≤ π)

#»v · #»w = 0 means θ is

right (θ = π/2)

Definition#»v and #»w are orthogonal if #»v · #»w = 0

Example

Are #»v = 〈1,√

2, 1, 0〉 and #»w = 〈1,−√

2, 1, 1〉 orthogonal? Yes,since

#»v · #»w = (1)(1) + (√

2)(−√

2) + (1)(1) + (0)(1) = 1− 2 + 1 + 0 = 0

The Dot ProductAngles

TheoremLet θ be the angle between two vectors #»v 6= #»

O and #»w 6= #»

O . Then#»v · #»w > 0 means θ is acute (0 ≤ θ < π/2)#»v · #»w < 0 means θ is

obtuse (π/2 < θ ≤ π)#»v · #»w = 0 means θ is

right (θ = π/2)

Definition#»v and #»w are orthogonal if #»v · #»w = 0

Example

Are #»v = 〈1,√

2, 1, 0〉 and #»w = 〈1,−√

2, 1, 1〉 orthogonal? Yes,since

#»v · #»w = (1)(1) + (√

2)(−√

2) + (1)(1) + (0)(1) = 1− 2 + 1 + 0 = 0

The Dot ProductAngles

TheoremLet θ be the angle between two vectors #»v 6= #»

O and #»w 6= #»

O . Then#»v · #»w > 0 means θ is acute (0 ≤ θ < π/2)#»v · #»w < 0 means θ is obtuse (π/2 < θ ≤ π)

#»v · #»w = 0 means θ is

right (θ = π/2)

Definition#»v and #»w are orthogonal if #»v · #»w = 0

Example

Are #»v = 〈1,√

2, 1, 0〉 and #»w = 〈1,−√

2, 1, 1〉 orthogonal? Yes,since

#»v · #»w = (1)(1) + (√

2)(−√

2) + (1)(1) + (0)(1) = 1− 2 + 1 + 0 = 0

The Dot ProductAngles

TheoremLet θ be the angle between two vectors #»v 6= #»

O and #»w 6= #»

O . Then#»v · #»w > 0 means θ is acute (0 ≤ θ < π/2)#»v · #»w < 0 means θ is obtuse (π/2 < θ ≤ π)#»v · #»w = 0 means θ is

right (θ = π/2)

Definition#»v and #»w are orthogonal if #»v · #»w = 0

Example

Are #»v = 〈1,√

2, 1, 0〉 and #»w = 〈1,−√

2, 1, 1〉 orthogonal? Yes,since

#»v · #»w = (1)(1) + (√

2)(−√

2) + (1)(1) + (0)(1) = 1− 2 + 1 + 0 = 0

The Dot ProductAngles

TheoremLet θ be the angle between two vectors #»v 6= #»

O and #»w 6= #»

O . Then#»v · #»w > 0 means θ is acute (0 ≤ θ < π/2)#»v · #»w < 0 means θ is obtuse (π/2 < θ ≤ π)#»v · #»w = 0 means θ is right (θ = π/2)

Definition#»v and #»w are orthogonal if #»v · #»w = 0

Example

Are #»v = 〈1,√

2, 1, 0〉 and #»w = 〈1,−√

2, 1, 1〉 orthogonal? Yes,since

#»v · #»w = (1)(1) + (√

2)(−√

2) + (1)(1) + (0)(1) = 1− 2 + 1 + 0 = 0

The Dot ProductAngles

TheoremLet θ be the angle between two vectors #»v 6= #»

O and #»w 6= #»

O . Then#»v · #»w > 0 means θ is acute (0 ≤ θ < π/2)#»v · #»w < 0 means θ is obtuse (π/2 < θ ≤ π)#»v · #»w = 0 means θ is right (θ = π/2)

Definition#»v and #»w are orthogonal if #»v · #»w = 0

Example

Are #»v = 〈1,√

2, 1, 0〉 and #»w = 〈1,−√

2, 1, 1〉 orthogonal? Yes,since

#»v · #»w = (1)(1) + (√

2)(−√

2) + (1)(1) + (0)(1) = 1− 2 + 1 + 0 = 0

The Dot ProductAngles

TheoremLet θ be the angle between two vectors #»v 6= #»

O and #»w 6= #»

O . Then#»v · #»w > 0 means θ is acute (0 ≤ θ < π/2)#»v · #»w < 0 means θ is obtuse (π/2 < θ ≤ π)#»v · #»w = 0 means θ is right (θ = π/2)

Definition#»v and #»w are orthogonal if #»v · #»w = 0

Example

Are #»v = 〈1,√

2, 1, 0〉 and #»w = 〈1,−√

2, 1, 1〉 orthogonal?

Yes,since

#»v · #»w = (1)(1) + (√

2)(−√

2) + (1)(1) + (0)(1) = 1− 2 + 1 + 0 = 0

The Dot ProductAngles

TheoremLet θ be the angle between two vectors #»v 6= #»

O and #»w 6= #»

O . Then#»v · #»w > 0 means θ is acute (0 ≤ θ < π/2)#»v · #»w < 0 means θ is obtuse (π/2 < θ ≤ π)#»v · #»w = 0 means θ is right (θ = π/2)

Definition#»v and #»w are orthogonal if #»v · #»w = 0

Example

Are #»v = 〈1,√

2, 1, 0〉 and #»w = 〈1,−√

2, 1, 1〉 orthogonal? Yes,since

#»v · #»w = (1)(1) + (√

2)(−√

2) + (1)(1) + (0)(1) = 1− 2 + 1 + 0 = 0

The Dot ProductAngles

ProblemDescribe all vectors orthogonal to #»v = 〈1, −3, 8〉 .

The vectors orthogonal to #»v are the vectors #»w = 〈w1, w2, w3〉satisfying #»v · #»w = 0. This gives the equation

0 = #»v · #»w = 〈1, −3, 8〉 · 〈w1, w2, w3〉 = w1 − 3w2 + 8w3

Solving this equation for w1 gives w1 = 3w2 − 8w3. The vectorsorthogonal to #»v are thus given by

#»w =

w1

w2

w3

=

3w2 − 8w3

w2

w3

= w2

310

+ w3

−801

where w2 and w3 can be chosen “freely.” The vectors orthogonalto #»v are the linear combinations of 〈3, 1, 0〉 and 〈−8, 0, 1〉 .

The Dot ProductAngles

ProblemDescribe all vectors orthogonal to #»v = 〈1, −3, 8〉 .

The vectors orthogonal to #»v are the vectors #»w = 〈w1, w2, w3〉satisfying #»v · #»w =

0. This gives the equation

0 = #»v · #»w = 〈1, −3, 8〉 · 〈w1, w2, w3〉 = w1 − 3w2 + 8w3

Solving this equation for w1 gives w1 = 3w2 − 8w3. The vectorsorthogonal to #»v are thus given by

#»w =

w1

w2

w3

=

3w2 − 8w3

w2

w3

= w2

310

+ w3

−801

where w2 and w3 can be chosen “freely.” The vectors orthogonalto #»v are the linear combinations of 〈3, 1, 0〉 and 〈−8, 0, 1〉 .

The Dot ProductAngles

ProblemDescribe all vectors orthogonal to #»v = 〈1, −3, 8〉 .

The vectors orthogonal to #»v are the vectors #»w = 〈w1, w2, w3〉satisfying #»v · #»w = 0.

This gives the equation

0 = #»v · #»w = 〈1, −3, 8〉 · 〈w1, w2, w3〉 = w1 − 3w2 + 8w3

Solving this equation for w1 gives w1 = 3w2 − 8w3. The vectorsorthogonal to #»v are thus given by

#»w =

w1

w2

w3

=

3w2 − 8w3

w2

w3

= w2

310

+ w3

−801

where w2 and w3 can be chosen “freely.” The vectors orthogonalto #»v are the linear combinations of 〈3, 1, 0〉 and 〈−8, 0, 1〉 .

The Dot ProductAngles

ProblemDescribe all vectors orthogonal to #»v = 〈1, −3, 8〉 .

The vectors orthogonal to #»v are the vectors #»w = 〈w1, w2, w3〉satisfying #»v · #»w = 0. This gives the equation

0 = #»v · #»w = 〈1, −3, 8〉 · 〈w1, w2, w3〉 = w1 − 3w2 + 8w3

Solving this equation for w1 gives w1 = 3w2 − 8w3. The vectorsorthogonal to #»v are thus given by

#»w =

w1

w2

w3

=

3w2 − 8w3

w2

w3

= w2

310

+ w3

−801

where w2 and w3 can be chosen “freely.” The vectors orthogonalto #»v are the linear combinations of 〈3, 1, 0〉 and 〈−8, 0, 1〉 .

The Dot ProductAngles

ProblemDescribe all vectors orthogonal to #»v = 〈1, −3, 8〉 .

The vectors orthogonal to #»v are the vectors #»w = 〈w1, w2, w3〉satisfying #»v · #»w = 0. This gives the equation

0 = #»v · #»w = 〈1, −3, 8〉 · 〈w1, w2, w3〉 = w1 − 3w2 + 8w3

Solving this equation for w1 gives w1 =

3w2 − 8w3. The vectorsorthogonal to #»v are thus given by

#»w =

w1

w2

w3

=

3w2 − 8w3

w2

w3

= w2

310

+ w3

−801

where w2 and w3 can be chosen “freely.” The vectors orthogonalto #»v are the linear combinations of 〈3, 1, 0〉 and 〈−8, 0, 1〉 .

The Dot ProductAngles

ProblemDescribe all vectors orthogonal to #»v = 〈1, −3, 8〉 .

The vectors orthogonal to #»v are the vectors #»w = 〈w1, w2, w3〉satisfying #»v · #»w = 0. This gives the equation

0 = #»v · #»w = 〈1, −3, 8〉 · 〈w1, w2, w3〉 = w1 − 3w2 + 8w3

Solving this equation for w1 gives w1 = 3w2 − 8w3.

The vectorsorthogonal to #»v are thus given by

#»w =

w1

w2

w3

=

3w2 − 8w3

w2

w3

= w2

310

+ w3

−801

where w2 and w3 can be chosen “freely.” The vectors orthogonalto #»v are the linear combinations of 〈3, 1, 0〉 and 〈−8, 0, 1〉 .

The Dot ProductAngles

ProblemDescribe all vectors orthogonal to #»v = 〈1, −3, 8〉 .

The vectors orthogonal to #»v are the vectors #»w = 〈w1, w2, w3〉satisfying #»v · #»w = 0. This gives the equation

0 = #»v · #»w = 〈1, −3, 8〉 · 〈w1, w2, w3〉 = w1 − 3w2 + 8w3

Solving this equation for w1 gives w1 = 3w2 − 8w3. The vectorsorthogonal to #»v are thus given by

#»w =

w1

w2

w3

=

3w2 − 8w3

w2

w3

= w2

310

+ w3

−801

where w2 and w3 can be chosen “freely.” The vectors orthogonalto #»v are the linear combinations of 〈3, 1, 0〉 and 〈−8, 0, 1〉 .

The Dot ProductAngles

ProblemDescribe all vectors orthogonal to #»v = 〈1, −3, 8〉 .

The vectors orthogonal to #»v are the vectors #»w = 〈w1, w2, w3〉satisfying #»v · #»w = 0. This gives the equation

0 = #»v · #»w = 〈1, −3, 8〉 · 〈w1, w2, w3〉 = w1 − 3w2 + 8w3

Solving this equation for w1 gives w1 = 3w2 − 8w3. The vectorsorthogonal to #»v are thus given by

#»w =

w1

w2

w3

=

3w2 − 8w3

w2

w3

=

w2

310

+ w3

−801

where w2 and w3 can be chosen “freely.” The vectors orthogonalto #»v are the linear combinations of 〈3, 1, 0〉 and 〈−8, 0, 1〉 .

The Dot ProductAngles

ProblemDescribe all vectors orthogonal to #»v = 〈1, −3, 8〉 .

The vectors orthogonal to #»v are the vectors #»w = 〈w1, w2, w3〉satisfying #»v · #»w = 0. This gives the equation

0 = #»v · #»w = 〈1, −3, 8〉 · 〈w1, w2, w3〉 = w1 − 3w2 + 8w3

Solving this equation for w1 gives w1 = 3w2 − 8w3. The vectorsorthogonal to #»v are thus given by

#»w =

w1

w2

w3

=

3w2 − 8w3

w2

w3

= w2

310

+ w3

−801

where w2 and w3 can be chosen “freely.” The vectors orthogonalto #»v are the linear combinations of 〈3, 1, 0〉 and 〈−8, 0, 1〉 .

The Dot ProductAngles

ProblemDescribe all vectors orthogonal to #»v = 〈1, −3, 8〉 .

The vectors orthogonal to #»v are the vectors #»w = 〈w1, w2, w3〉satisfying #»v · #»w = 0. This gives the equation

0 = #»v · #»w = 〈1, −3, 8〉 · 〈w1, w2, w3〉 = w1 − 3w2 + 8w3

Solving this equation for w1 gives w1 = 3w2 − 8w3. The vectorsorthogonal to #»v are thus given by

#»w =

w1

w2

w3

=

3w2 − 8w3

w2

w3

= w2

310

+ w3

−801

where w2 and w3 can be chosen “freely.”

The vectors orthogonalto #»v are the linear combinations of 〈3, 1, 0〉 and 〈−8, 0, 1〉 .

The Dot ProductAngles

ProblemDescribe all vectors orthogonal to #»v = 〈1, −3, 8〉 .

The vectors orthogonal to #»v are the vectors #»w = 〈w1, w2, w3〉satisfying #»v · #»w = 0. This gives the equation

0 = #»v · #»w = 〈1, −3, 8〉 · 〈w1, w2, w3〉 = w1 − 3w2 + 8w3

Solving this equation for w1 gives w1 = 3w2 − 8w3. The vectorsorthogonal to #»v are thus given by

#»w =

w1

w2

w3

=

3w2 − 8w3

w2

w3

= w2

310

+ w3

−801

where w2 and w3 can be chosen “freely.” The vectors orthogonalto #»v are the linear combinations of 〈3, 1, 0〉 and 〈−8, 0, 1〉 .