7/28/2019 Analytic Geometry Summary Sheet

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Mr. Simonds MTH 253

V e c t o r B a s i c s s t u f f t o b e m e m o r i z e d | 1

Basic Vector Language and Vector Arithmetic

Fact 1: The three standard unit vectors are 1,0,0i = , 0,1,0j = and 0,0,1k = . The

vector1 2 3, ,u u u u=

G

can also be written as 1 2 3 u u i u j u k = + +

G

. Consequently, the

components of the vector u

G

; 1u , 2u , and 3u ; are called, respectively, the i component,the j component, and the k component.

Fact 2: To add or subtract 2 vectors you add or subtract the corresponding components of the

two vectors. To multiply a vector by a scalar (number), you multiply each component of

the vector by the scalar.

Fact 3: Two non-zero vectors are called parallel if and only if they point in exactly the same

direction or they point in exactly opposite directions.

Fact 4: A non-zero vector is said to be parallel to a line if and only if the vector couldbe drawn

entirely along the line.

Fact 5: A non-zero vector is said to be parallel to a plane if and only if the vector couldbe drawn

entirely on the plane.

Fact 6: A non-zero vector is said to be perpendicular (or normal) to a plane if and only if the

vector forms a right angle when drawn tail-to-tail with any vector parallel to the plane.

Vector Norms

Def 1: The norm of the vector 1 2 3, ,u u u u=G

is 2 2 21 2 3u u u u= + +G

. In 2-dimensions and 3-

dimenstions this quantity is commonly referred to as the length of the vector. In appliedproblems this quantity is commonly referred to as the magnitude of the vector.

Fact 7: A vector, vG

, is called a unit vector if and only if 1v =G

. Unit vectors are generally

annotated thusly: v .

Fact 8: For a non-zero vector vG

, v

uv

=

G

Gis the unit vector that points in the same direction as

vG

. The process of dividing a vector by its length is called normalizing the vector.

The Dot Product

Def 2: If 1 2 3, ,u u u u=G

and 1 2 3, ,v v v v=G

, then 1 1 2 2 3 3u v u v u v u v = + +G G

.

Fact 9: The smallest angle, , formed by non-zero vectors uG

and vG

when uG

and vG

are drawn

tail-to-tail satisfies the equation ( )cosu v

u v

=

G G

G G.

7/28/2019 Analytic Geometry Summary Sheet

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Mr. Simonds MTH 253

2 | V e c t o r B a s i c s s t u f f t o b e m e m o r i z e d

The Cross Product

Def 3: If1 2 3, ,u u u u=

G

and1 2 3, ,v v v v=

G

, then

( ) ( ) ( )1 2 3 2 3 3 2 1 3 3 1 1 2 2 11 2 3

i j k

u v u u u u v u v i u v u v j u v u v k

v v v

= = + G G

.

Fact 10: ( )u v v u = G G G G

Fact 11: For two non-zero, non-parallel vectors uG

and vG

, u vG G

is perpendicular to any plane that

is parallel to both uG

and vG

. That is, ( )u v u G G G

and ( )u v v G G G

.

Vector Projections

Def 4 and 5: compuv u

vu

=G

G G

G

Gand ( )proj compu u

v uu uv v

u u u

= = G G

G GG G

G G

G G G

Fact 12: comp 0 proju uv v u> G GG G G

and comp 0 proju uv v u< G GG G G

Additional Fundamental Facts about Vectors, Lines, and Planes

Fact 13: Two non-zero vectors uG

and vG

form a right angle when drawn tail-to-tail if and only if

0u v =G G

. Two such vectors are said to be perpendicular or orthogonal.

Fact 14: Two vectors uG

and vG

are parallel if and only if u k v=G G

for some non-zero scalar k.

Fact 15: A line that is parallel to the vector1 2 3, ,u u u u=

G

and passes through the point

( )0 0 0, ,x y z can be modeled by the vector function ( ) 0 1 0 2 0 3, ,r t x u t y u t z u t = + + +G

and the symmetric equations 0 0 0

1 2 3

x x y y z z

u u u

= = . The vector u

G

is called a

direction vector for the line.

Fact 16: A plane that is perpendicular to the vector 1 2 3, ,u u u u=G

and passes through the point

( )0 0 0, ,x y z can be modeled by the equation ( ) ( ) ( )1 0 2 0 3 0 0u x x u y y u z z + + = .The vector u

G

is called a normal vector for the plane.

Fact 17: The vector1 2 3, ,u u u u=

G

is a normal vector for any plane with an equation of form

1 2 3u x u y u z k + + = .