Lesson 2: Vectors and the Dot Product
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Transcript of Lesson 2: Vectors and the Dot Product
Section 9.2–3Vectors and the Dot Product
Math 21a
February 6, 2008
Announcements
I The MQC is open: Sun-Thu 8:30pm–10:30pm, SC 222I Homework for Friday 2/8:
I Section 9.2: 4, 6, 26, 33, 34I Section 9.3: 10, 18, 24, 25, 34I Section 9.4: 1*
Outline
VectorsAlgebra of VectorsComponentsStandard basis vectorsLength
The Dot ProductWorkConceptPropertiesUses
What is a vector?
Definition
I A vector is something that has magnitude and direction
I We denote vectors by boldface (v) or little arrows (~v). One isgood for print, one for script
I Given two points A and B in flatland or spaceland, the vectorwhich starts at A and ends at B is called the displacement
vector−→AB.
I Two vectors are equal if they have the same magnitude anddirection (they need not overlap)
A
B
v
C
D
u
Vector or scalar?
DefinitionA scalar is another name for a real number.
Example
Which of these are vectors or scalars?
(i) Cost of a theater ticket
scalar
(ii) The current in a river
vector
(iii) The initial flight path from Boston to New York
vector
(iv) The population of the world
scalar
Vector or scalar?
DefinitionA scalar is another name for a real number.
Example
Which of these are vectors or scalars?
(i) Cost of a theater ticket scalar
(ii) The current in a river vector
(iii) The initial flight path from Boston to New York vector
(iv) The population of the world scalar
Vector addition
DefinitionIf u and v are vectors positioned so the initial point of v is theterminal point of u, the sum u + v is the vector whose initial pointis the initial point of u and whose terminal point is the terminalpoint of v.
u
vu + v
The triangle law
u
v
u
v
u + v
The parallelogram law
Opposite and difference
DefinitionGiven vectors u and v,
I the opposite of v is the vector −v that has the same lengthas v but points in the opposite direction
I the difference u− v is the sum u + (−v)
u
v
−v
u− v
Scaling vectors
DefinitionIf c is a nonzero scalar and v is a vector, the scalar multiple cv isthe vector whose
I length is |c | times the length of v
I direction is the same as v if c > 0 and opposite v if c < 0
If c = 0, cv = 0.
v
2v
−12v
Scaling vectors
DefinitionIf c is a nonzero scalar and v is a vector, the scalar multiple cv isthe vector whose
I length is |c | times the length of v
I direction is the same as v if c > 0 and opposite v if c < 0
If c = 0, cv = 0.
v
2v
−12v
Scaling vectors
DefinitionIf c is a nonzero scalar and v is a vector, the scalar multiple cv isthe vector whose
I length is |c | times the length of v
I direction is the same as v if c > 0 and opposite v if c < 0
If c = 0, cv = 0.
v
2v
−12v
Properties
TheoremGiven vectors a, b, and c and scalars c and d, we have
1. a + b = b + a
2. a + (b + c) = (a + b) + c
3. a + 0 = a
4. a + (−a) = 0
5. c(a + b) = ca + cb
6. (c + d)a = ca + da
7. (cd)a = c(da)
8. 1a = a
These can be verified geometrically.
Components defined
Definition
I Given a vector a, it’s often useful to move the tail to O andmeasure the coordinates of the head. These are called thecomponents of a, and we write them like this:
a = 〈a1, a2, a3〉
or just two components if the vector is the plane. Note theangle brackets!
I Given a point P in the plane or space, the position vector of
P is the vector−→OP.
FactGiven points A(x1, y1, z1) and B(x2, y2, z2) in space, the vector
−→AB
has components
−→AB = 〈x2 − x1, y2 − y1, z2 − z1〉
Components defined
Definition
I Given a vector a, it’s often useful to move the tail to O andmeasure the coordinates of the head. These are called thecomponents of a, and we write them like this:
a = 〈a1, a2, a3〉
or just two components if the vector is the plane. Note theangle brackets!
I Given a point P in the plane or space, the position vector of
P is the vector−→OP.
FactGiven points A(x1, y1, z1) and B(x2, y2, z2) in space, the vector
−→AB
has components
−→AB = 〈x2 − x1, y2 − y1, z2 − z1〉
Components defined
Definition
I Given a vector a, it’s often useful to move the tail to O andmeasure the coordinates of the head. These are called thecomponents of a, and we write them like this:
a = 〈a1, a2, a3〉
or just two components if the vector is the plane. Note theangle brackets!
I Given a point P in the plane or space, the position vector of
P is the vector−→OP.
FactGiven points A(x1, y1, z1) and B(x2, y2, z2) in space, the vector
−→AB
has components
−→AB = 〈x2 − x1, y2 − y1, z2 − z1〉
Vector algebra in components
TheoremIf a = 〈a1, a2, a3〉 and b = 〈b1, b2, b3〉, and c is a scalar, then
I a + b = 〈a1 + b1, a2 + b2, a3 + b3〉I a− b = 〈a1 − b1, a2 − b2, a3 − b3〉I ca = 〈ca1, ca2, ca3〉
Useful vectors
DefinitionWe define the standard basis vectors i = 〈1, 0, 0〉, j = 〈0, 1, 0〉,k = 〈0, 0, 1〉. In script, they’re often written as ı̂, ̂, k̂.
FactAny vector a can be written as a linear combination of thestandard basis vectors
〈a1, a2, a3〉 = a1i + a2j + a3k.
Useful vectors
DefinitionWe define the standard basis vectors i = 〈1, 0, 0〉, j = 〈0, 1, 0〉,k = 〈0, 0, 1〉. In script, they’re often written as ı̂, ̂, k̂.
FactAny vector a can be written as a linear combination of thestandard basis vectors
〈a1, a2, a3〉 = a1i + a2j + a3k.
Length
DefinitionGiven a vector v, its length is the distance between its initial andterminal points.
FactThe length of a vector is the square root of the sum of the squaresof its components:
|〈a1, a2, a3〉| =√
a21 + a2
2 + a23
Length
DefinitionGiven a vector v, its length is the distance between its initial andterminal points.
FactThe length of a vector is the square root of the sum of the squaresof its components:
|〈a1, a2, a3〉| =√
a21 + a2
2 + a23
Early vector users
I Caspar Wessel (Norwegian and Danish, 1745–1818)
I Jean Robert Argand (French 1768–1822),
I Carl Friedrich Gauss (German, 1777–1855)
I Sir William Rowan Hamilton (Irish, 1805–1865)
Outline
VectorsAlgebra of VectorsComponentsStandard basis vectorsLength
The Dot ProductWorkConceptPropertiesUses
DefinitionWork is the energy needed to move an object by a force.
If the force is expressed as a vector F and the displacement avector D, the work is
W = |F| |D| cos θ
where θ is the angle between the vectors.
θ D
F
Work is |F| times this distance
DefinitionWork is the energy needed to move an object by a force.
If the force is expressed as a vector F and the displacement avector D, the work is
W = |F| |D| cos θ
where θ is the angle between the vectors.
θ D
F
Work is |F| times this distance
DefinitionIf a and b are any two vectors in the plane or in space, the dotproduct (or scalar product) between them is the quantity
a · b = |a| |b| cos θ,
where θ is the angle between them.
Another way to say this is that a · b is |b| times the length of theprojection of a onto b.
a
ba · b is |b| times this length
DefinitionIf a and b are any two vectors in the plane or in space, the dotproduct (or scalar product) between them is the quantity
a · b = |a| |b| cos θ,
where θ is the angle between them.Another way to say this is that a · b is |b| times the length of theprojection of a onto b.
a
ba · b is |b| times this length
Geometric properties of the dot product
Fact
I Two vectors are perpendicular or orthogonal if their dot
product is zero (i.e., cos θ = 90◦ =π
2)
I The law of cosines can be expressed as
|a + b|2 = |a|2 + |b|2 − 2 |a| |b| cos θ
= |a|2 + |b|2 − 2a · b
I In components, if a = 〈a1, a2, a3〉 and b = 〈b1, b2, b3〉, then
a · b = a1b1 + a2b2 + a3b3
Geometric properties of the dot product
Fact
I Two vectors are perpendicular or orthogonal if their dot
product is zero (i.e., cos θ = 90◦ =π
2)
I The law of cosines can be expressed as
|a + b|2 = |a|2 + |b|2 − 2 |a| |b| cos θ
= |a|2 + |b|2 − 2a · b
I In components, if a = 〈a1, a2, a3〉 and b = 〈b1, b2, b3〉, then
a · b = a1b1 + a2b2 + a3b3
Geometric properties of the dot product
Fact
I Two vectors are perpendicular or orthogonal if their dot
product is zero (i.e., cos θ = 90◦ =π
2)
I The law of cosines can be expressed as
|a + b|2 = |a|2 + |b|2 − 2 |a| |b| cos θ
= |a|2 + |b|2 − 2a · b
I In components, if a = 〈a1, a2, a3〉 and b = 〈b1, b2, b3〉, then
a · b = a1b1 + a2b2 + a3b3
More geometric properties of the dot product
FactThe angle between two nonzero vectors a and b is given by
cos θ =a · b|a| |b|
,
where θ is taken to be between 0 and π.
FactThe angle between two nonzero vectors a and b is
I acute if a · b > 0
I obtuse if a · b < 0
I right if a · b = 0;
The vectors are parallel if a · b = ± |a| |b|.
I b is a positive multiple of a if a · b = |a| |b|I b is a negative multiple of a if a · b = − |a| |b|
More geometric properties of the dot product
FactThe angle between two nonzero vectors a and b is given by
cos θ =a · b|a| |b|
,
where θ is taken to be between 0 and π.
FactThe angle between two nonzero vectors a and b is
I acute if a · b > 0
I obtuse if a · b < 0
I right if a · b = 0;
The vectors are parallel if a · b = ± |a| |b|.
I b is a positive multiple of a if a · b = |a| |b|I b is a negative multiple of a if a · b = − |a| |b|
More geometric properties of the dot product
FactThe angle between two nonzero vectors a and b is given by
cos θ =a · b|a| |b|
,
where θ is taken to be between 0 and π.
FactThe angle between two nonzero vectors a and b is
I acute if a · b > 0
I obtuse if a · b < 0
I right if a · b = 0;
The vectors are parallel if a · b = ± |a| |b|.
I b is a positive multiple of a if a · b = |a| |b|I b is a negative multiple of a if a · b = − |a| |b|
More geometric properties of the dot product
FactThe angle between two nonzero vectors a and b is given by
cos θ =a · b|a| |b|
,
where θ is taken to be between 0 and π.
FactThe angle between two nonzero vectors a and b is
I acute if a · b > 0
I obtuse if a · b < 0
I right if a · b = 0;
The vectors are parallel if a · b = ± |a| |b|.
I b is a positive multiple of a if a · b = |a| |b|I b is a negative multiple of a if a · b = − |a| |b|
More geometric properties of the dot product
FactThe angle between two nonzero vectors a and b is given by
cos θ =a · b|a| |b|
,
where θ is taken to be between 0 and π.
FactThe angle between two nonzero vectors a and b is
I acute if a · b > 0
I obtuse if a · b < 0
I right if a · b = 0;
The vectors are parallel if a · b = ± |a| |b|.I b is a positive multiple of a if a · b = |a| |b|
I b is a negative multiple of a if a · b = − |a| |b|
More geometric properties of the dot product
FactThe angle between two nonzero vectors a and b is given by
cos θ =a · b|a| |b|
,
where θ is taken to be between 0 and π.
FactThe angle between two nonzero vectors a and b is
I acute if a · b > 0
I obtuse if a · b < 0
I right if a · b = 0;
The vectors are parallel if a · b = ± |a| |b|.I b is a positive multiple of a if a · b = |a| |b|I b is a negative multiple of a if a · b = − |a| |b|
Examples
Example
Find the sum of the following pairs of vectors geometrically andalgebraically.
(i) a = 〈3,−1〉 and b = 〈−2, 4〉(ii) a = 〈0, 1, 2〉 and b = 〈0, 0,−3〉
What is the angle between the two vectors in each case?
Solution
(i) a + b = 〈1, 3〉, |a| =√
10, |b| =√
20. So
cos θ =a · b|a| |b|
=−6− 4√
10√
20=
−10√10√
20= − 1√
2=⇒ θ =
3π
4
(ii) a + b = 〈0, 1,−1〉, while
cos θ =0 + 0− 6√
5√
9= − 2√
5
Examples
Example
Find the sum of the following pairs of vectors geometrically andalgebraically.
(i) a = 〈3,−1〉 and b = 〈−2, 4〉(ii) a = 〈0, 1, 2〉 and b = 〈0, 0,−3〉
What is the angle between the two vectors in each case?
Solution
(i) a + b = 〈1, 3〉, |a| =√
10, |b| =√
20. So
cos θ =a · b|a| |b|
=−6− 4√
10√
20=
−10√10√
20= − 1√
2=⇒ θ =
3π
4
(ii) a + b = 〈0, 1,−1〉, while
cos θ =0 + 0− 6√
5√
9= − 2√
5
Properties
FactIf a, b and c are vectors are c is a scalar, then
1. a · a = |a|22. a · b = b · a3. a · (b + c) = a · b + a · c
4. (ca) ·b = c(a ·b) = a · (cb)
5. 0 · a = 0
Example
The dot product can be used to measure how similar two vectorsare. Consider it a compatibility index. If two vectors point inapproximately the same direction, we get a positive dot product. Iftwo vectors are orthogonal, we get a zero dot product. If twovectors point in approximately opposite directions, we get anegative dot product.Consider the following categories,
1. Football
2. Sushi
3. Classical music
Now create a vector in R3 rating your preference in each categoryfrom −5 to 5, where −5 expresses extreme dislike and 5 expressesadoration. Dot your vector with your neighbor’s.
Example
Fifi, a poodle, drags her owner along a sidewalk that is 200 meterslong. If Fifi exerts a force of two newtons on the leash, and theleash is at an angle 45◦ from the ground, how much work does Fifido?