Vectors

Scalars & Vectors

• Vectors– Quantity with both

magnitude & direction– Does NOT follow

elementary arithmetic/algebra rules

– Examples – position, force, moment, velocities, acceleration

Tail

Head

Line of Action

Direction/Angle

Magnitude

Parallelogram Law

• The resultant of two forces can be obtained by– Joining the vectors at their

tailsA

B

A+B

Constructing a parallelogram

The resultant is the diagonal of the parallelogram

Triangle Construction

• The resultant of two forces can be obtained by– Joining the vectors in

tip-to-tail fashionA B

R The resultant extends

from the tail of A to

the head of the B

Vector Addition

• Does

A + B = B + A ?

A BR

AB

R

YES! - commutative

Vector Subtraction

A - B = A + (-B)

A B

R

-B

A

-B

Vector Subtraction

• Does

A – B = B - A ?

NO! – opposite sense

RA

-B

-R -A

B

Vector Operations

• Multiplication & Division of Vector (A) by Scalar (a)

a * A = aA

A 2A

A -.5A

2 * A = 2A

-.5 * A = -.5A

Representation of a Vector

Given the points and , the vector a with representation is

a

1 1 1( , , )A x y z2 2 2( , , )B x y z

Find the vector represented by the directed line segment with initial point A(2,-3,4) and terminal point B(-2,1,1).

AB��������������

2 1 2 1 2 1, ,x x y y z z

2 2,1 ( 3),1 4a

4,4, 3a

Magnitude of a vector

Determine the magnitude of the following:

ExampleIf 4,0,3 and 2,1,5 , find a and the vectors a+b, a-b, 3b,and 2a+5b.a b

2 2 24 0 3 25 5a 4,0,3 2,1,5

4 2,0 1,3 5

2,1,8

a b

a b

a b

4,0,3 2,1,5

4 ( 2),0 1,3 5

6, 1, 2

a b

a b

a b

3 3 2,1,5 3( 2),3(1),3(5) 6,3,15b

2 5 2 4,0,3 5 2,1,5

2 5 8,0,6 10,5,25

2 5 2,5,31

a b

a b

a b

Parallel • Two vectors are parallel to each other if one is the scalar multiple of the other.

Determine if the two vectors are parallel

These are parallel since

b= -3a

These are not parallel since 4(1/2) =2 , but 10(1/2)=5 not -9

Unit vectors

i= 1,0,0 j= 0,1,0 k= 0,0,1

Any vector that has a magnitude of 1 is considered a unit vector.

Can you think of a unit vector?

Standard Basis Vectors

1 2 3

1 2 3 1 2 3

If a= , , , then we can write

a= , , ,0,0 0, ,0 0,0,

a a a

a a a a a a

1 2 3a= 1,0,0 0,1,0 0,0,1a a a

1 2 3a=a i a j a k Example- Write in terms of the standard basis vector i,j,k. 1, 2,6

1, 2,6 i - 2 j 6k

Example

If a = i + 2j - 3k and b = 4i + 7k, express the vector 2a+3b in terms of i,j,k.

2a+3b=2(i + 2j - 3k)+3(4i + 7k)

2a+3b=2i + 4j - 6k+ 12i + 21k

2a+3b=14i+4j+15k

Unit Vectors

1 au = a =

a a

1u= a

31

u= (2i - j - 2k)3

The unit vector in the same direction of a is 1 a

u = a =a a

Find a unit vector in the same direction as 2i – j – 2k.

We are looking for a vector in the same direction as the original vector, but is also a unit vector.

Let’s first find the magnitude 2 2 22i - j - 2k 2 ( 1) ( 2) 9 3

2 1 2u= i - j - k

3 3 3

Check?

Same direction?

Magnitude = 1?

Homework

• P649– 4,5,7,9,11,15,17,19