Vectors in 2 and 3 dimensions

Definition:

A scalar has magnitude

A vector has magnitude and direction

Representing vectors

A

B

AB =63

6

3a

a = = 6i + 3j

i =10

⎛

⎝⎜

⎞

⎠⎟ j =

01

⎛

⎝⎜

⎞

⎠⎟Unit Vectors

The magnitude of a vector

A

B

6

3a

ABu ruu

= 62 + 32

= 45

=3 5

3 Dimensions

• In 3 dimensions, we consider the x, y and z axis.

• As unit vectors, we use i, j, k

• E.g. 3

−25

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟=3i −2 j +5k

Magnitude of a 3D vector

• The formula for working out the magnitude (length) of a 3D vector is very similar to that for a 2D vector:

a

b

c

= a2 +b2 +c2

Finding Unit Vectors• A unit vector in the direction of v can be found as:

• A vector of length a in the direction of a vector, v, can be found as:

1

v• v

Finding Vectors of Given Lengths

a

v• v

Finding Unit Vectors

• Example: Find the unit vector in the direction of 5i -2j + 4k

Finding Unit Vectors

• Example: Find the vector of length in the direction of 5i -2j + 4k

5

Equal vectorsTwo vectors are equal if they have the same magnitude and direction.

All of the following vectors are equal:

They are the same length and parallel.

a

bc

d

ef

g

The negative of a vector

Here is the vector AB =52

a

A

B

Suppose the arrow went in the opposite direction:

A

B

We can describe this vector as:

BA –a–5–2

or

Multiplying vectors by scalars

The vector 2a has the same direction but is twice as long.

a =32

2a =64

a 2a

Adding vectors

If a =53

and b =3

–2

Find a + b

We can represent this addition in the following diagram:

a b

a + b

a + b =81

Subtracting vectors

and b =–23

a =44

a b

a – b

a – b =61

–b a –b

Subtracting vectors

and b =–23

a =44

a

a – 2b

a – 2b =44

–2bb

-23

44

4-6

8-2

- 2 = + =

O

KJ

IHGF

EDC

L M

b

a

Vectors on a tangramA tangram is an ancient Chinese puzzle in which a square ABCD is divided as follows:

A B

CD

F

E

G

H

I

J

Suppose,

AE = a and AF = b

Write the following in terms of a and b.

FC =

b

a

3b

HJ = –a

IG = a – 2b

CB = 2a – 4b

HI = b – a

HD = 2b – 3a

b

O

C

B

C is ¼ of the way along OB

OC =

BC =

1

4b

3

4b

s

OC

B

C divides the line OB in the ratio 1:5

OC =

BC =

1

6b

5

6b

a

b

O A

C

B

C divides the line AB in the ratio 1:2

AB = -a + b

AC = 1/3 (–a + b)

OC = 2 1

3 3a b

Additional points to note:

• A position vector is one that starts from the origin, such as

• If one vector is a multiple of the other, the vectors are parallel

OAu ruu