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MTH 254: Important Vector Formulas for Test 1, Exams of Calculus

Portland Community CollegeCalculus

Essential formulas and facts for the mth 254 test, focusing on vector arithmetic, norms, the dot product, the cross product, and vector functions. It covers topics such as unit vectors, parallel and perpendicular vectors, lines, planes, and velocity and acceleration.

Typology: Exams

Pre 2010

Uploaded on 08/19/2009

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koofers-user-axv 🇺🇸

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MTH 254 - Important Formulas for Test 1
The following facts and formulae are things you need to memorize. The facts and formulae
that appear on pages 1 and 2 were covered in MTH 253. All statements are in reference to 3-
dimensional vectors.
Basic Vector Language and Vector Arithmetic
The three standard unit vectors are ˆ1, 0, 0i=, ˆ0,1,0j= and ˆ0, 0,1k=. The vector
123
,,uuuu=
G can also be written as 12 3
ˆ
ˆˆ
uuiujuk=+ +
G
. Consequently, the components of the
vector u
G; 1
u, 2
u, and 3
u; are called, respectively, the ˆ
i
−
component, the ˆ
j
−component, and the
ˆ
k−component.
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.
Two non-zero vectors are called parallel if and only if they both
could
each be drawn in their
entirety along the same line. That is, two non-zero are parallel if and only if they point in exactly
the same direction or they point in exactly opposite directions.
A non-zero vector is said to be parallel to a line if and only if the vector
could
be drawn entirely
along the line.
A non-zero vector is said to be parallel to a plane if and only if the vector
could
be drawn entirely
on the plane.
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
The norm of the vector 123
,,uuuu=
G is 222
123
uuuu=++
G
. In 2-dimensions and 3-
dimenstions this quantity is commonly referred to as the length of the vector. In applied problems
this quantity is commonly referred to as the magnitude of the vector.
Fact: A vector,
v
G, is called a unit vector if and only if 1v
=
G
. Unit vectors are generally
annotated thusly: ˆ
v.
Fact: For a non-zero vector v
G, the vector ˆv
uv
=
G
G
is the unit vector that points in the same
direction as v
G. The process of dividing a vector by its length is called normalizing the
vector.
pf3

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MTH 254 - Important Formulas for Test 1

The following facts and formulae are things you need to memorize. The facts and formulae

that appear on pages 1 and 2 were covered in MTH 253. All statements are in reference to 3-

dimensional vectors.

Basic Vector Language and Vector Arithmetic

The three standard unit vectors are

i = 1,0,0 ,

j = 0,1,0 and

k = 0,0,1. The vector

1 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

;

1

u ,

2

u , and

3

u ; are called, respectively, the

i − component, the

j − component, and the

k − component.

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.

Two non-zero vectors are called parallel if and only if they bothcould each be drawn in their

entirety along the same line. That is, two non-zero are parallel if and only if they point in exactly

the same direction or they point in exactly opposite directions.

A non-zero vector is said to be parallel to a line if and only if the vectorcould be drawn entirely

along the line.

A non-zero vector is said to be parallel to a plane if and only if the vectorcould be drawn entirely

on the plane.

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

The norm of the vector

1 2 3

u = u , u ,u

G

is

2 2 2

1 2 3

u = u + u +u

G

. In 2-dimensions and 3-

dimenstions this quantity is commonly referred to as the length of the vector. In applied problems

this quantity is commonly referred to as the magnitude of the vector.

Fact: A vector, v

G

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

G

. Unit vectors are generally

annotated thusly: vˆ.

Fact: For a non-zero vector v

G

, the vector ˆ

v

u

v

G

G

is the unit vector that points in the same

direction as v

G

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

vector.

MTH 254 Test 1 Material

The Dot Product

If

1 2 3

u = u , u ,u

G

and

1 2 3

v = v v, ,v

G

, then

1 1 2 2 3 3

u ⋅ v = u v + u v +u v

G G

.

Fact: The smallest angle, θ , formed by non-zero vectors u

G

and v

G

when u

G

and v

G

are drawn tail-

to-tail satisfies the equation ( )

cos

u v

u v

G G

G G

.

The Cross Product

If

1 2 3

u = u , u ,u

G

and

1 2 3

v = v v, ,v

G

, then

( ) ( ) ( )

1 2 3 2 3 3 2 1 3 3 1 1 2 2 1

1 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: ( )

u × v = − v ×u

G G G G

.

Fact: For two non-zero, non-parallel vectors u

G

and v

G

, u ×v

G G

is perpendicular to any plane that is

parallel to both u

G

and v

G

. That is, ( )

u × v ⊥u

G G G

and ( )

u × v ⊥v

G G G

.

Additional Fundamental Facts about Vectors, Lines, and Planes

Fact: Two non-zero vectors u

G

and v

G

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

u ⋅ v= 0

G G

. Two such vectors are said to be perpendicular.

Fact: Two vectors u

G

and v

G

are parallel if and only if u =k v

G G

for some non-zero scalar k.

Fact: A line that is parallel to the vector

1 2 3

u = u , u ,u

G

and passes through the point

( )

O O O

x y z can be modeled by the vector function ( )

1 2 3

O O O

r t = x + u t y + u t z +u t

G

.

The vector u

G

is called a direction vector for the line.

Fact: A plane that is perpendicular to the vector

1 2 3

u = u , u ,u

G

and passes through the point

( )

O O O

x y z can be modeled by the equation ( ) ( ) ( )

1 2 3

O O O

u x − x + u y − y + u z − z =.

The vector u

G

is called a normal vector for the plane.

Fact: The vector

1 2 3

u = u , u ,u

G

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

1 2 3

u x + u y + u z = k.