VectorsVectors are represented by a directed line segment its length representing the magnitude and an arrow indicating the direction

A

B

orAB��������������

u

u

This vector is named

CD��������������

v= =6

2

DC��������������

= -v =6

2

EF��������������

w ==4

3

FE��������������

= -w =4

3

These are also known as COLUMN VECTORS

C

Dv

E

F

w

C

Dv

E

F

w

CD��������������

v= =6

2

This is also known as the components of v

Q

Pa

b

It can be calculated using Pythagoras

If a

PQb

��������������

The magnitude of a vector is represented by its

length and is written or AB s��������������

2 2PQ a b ��������������

Example If7

find s1

s

s 2 2a b

2 27 (1) 50 5 2

Vectors are only equal if they have the samemagnitude and direction.

Equal Vectors

a

bu

cd

v

If u v then a c and b d

au

b

For vectors

andc

vd

cv

d

For vectors and

ew

f

Addition Of Vectors

then the resultant vector

v w c e

d f

v w

v w

AB��������������

=5

4

BC��������������

=3

1

AC��������������

i) Find the components of

AC ��������������

5 3

4 1

AB��������������

BC��������������

=

5 3

4 ( 1)

=8

3

The Zero Vector0

0

is called the zero vector written 0

0

0

AB��������������

BA��������������

5

3

5

3

If5

3

AB��������������

find the components of

AB��������������

BA��������������

Subtraction of Vectors

For vectors and if and a c

u v u vb d

a cu v

b d

Page 236 Exercise 13D questions 1 to 3

Multiplication by a scalar

If then and vector is parallel to a ka

u ku ku ub kb

u

ku

Hence if then u is parallel to u kv v

Conversely if is parallel to then uu v kv

Unit Vectors

For any vector v there exists a parallel vector u of magnitude 1 unit. This is called a Unit Vector.

i.e. Find the components of the unit vector u parallel to vector3

4v

2 23 4v

5

Since the magnitude of v is 5, the unit vector u must be 1/5 v

1

5u v

31

45

35

45

Position Vectors

where and are the position vectors of A and B.AB b a a b ��������������

If P and Q have coordinates (4,8) and (2,3) find the components of PQ��������������

PQ q p ��������������

2 4

3 8

2

5

Collinearity

We have seen that if a vector v = ku then v must be parallel to u.

If vectors v and u also have a point in common then because they are parallel they must lie on the same line so by definition must be collinear.

If where is a scalar then is parallel to BC.

If B is also a point in common to both AB and BC, then

A, B and C are said to be collinear.

AB k BC k AB��������������������������������������������������������

Prove that the points A(2,4), B(8,6) and C(11,7) are collinear.

AB b a ��������������

8 2

6 4

6

2

BC c b ��������������

11 8

7 6

3

1

2 AB BC����������������������������

is parallel to AB BC����������������������������

B is a point in common to both AB and BC so A, B and C are collinear

Section Formula

If p is the position vector of the point P that divides AB in the ratio m:n then:

n mp a b

m n m n

A

B

Pm

n

A and B have coordinates (3,2) and (7,14) respectively. Find the coordinates of the point P that divides AB in the ratio 1:3

1. Draw a quick sketch

(3,2)

(7,14)

P1

3

3 73 1

2 144 4p

3 713

2 144

9 71

6 144

4

5

the coordinates of P are (4,5)

3 Dimensional Vectors

x

y

z

A

The point A has a position relative to the x y and z axis

3

4

6

A(3, 4, 6)

Find the coordinates of P

y

x

z

P

4

2

1

P(4, 2, 1)

O

Find the coordinates of Q

y

x

z

Q

-1-2

-3

Q(-1, -2, -3)

3D Unit VectorsA vector can also be defined in terms of i, j and k where i, j and k are unit vectors in the x, y, and z directions respectively.

y

x

z

i 1

j 11 k

1

0

0

i

0

1

0

j

0

0

1

k

In component form the vectors are written as

Any vector can be expressed as a combination of its components.

3

4 can be written

2

v

3 4 2v i j k

Properties of 3D vectors

P

Q

42

5

4

2

5

PQ

��������������

2 2 2PQ PR QR

R

S2 2 2PS SR QR 2 2 24 2 5PQ

45

3 5

2 2 2If then

a

PQ b PQ a b c

c

����������������������������

2 2If then

a

PQ b PQ a b c

c

����������������������������

If , then

a d a d

u b v e u v b e

c f c f

Addition / Subtraction

If then k

a ka

u b u kb

c kc

Scalar

If (a, b, c) and (d, e, f) then R S RS s r ��������������

Position Vector

Section Formula

If A is (4, -6, 12) and B is (4, 4, -3). Find P that divides AB in the ratio 3:2��������������

A (4, -6, 12)

B (4, 4, -3)

P3

2

4 42 3

6 45 5

12 3

p

8 121

12 125

24 9

20 41

0 05

15 3

(4,0,3)p

The Scalar Product

a

b

For two vectors a and b, the scalar product is defined by

. cosa b a b

Where is the angle between a and b, 00 180

The scalar product is also known as the dot product.

The vectors must be directed away from the point of intersection.

0 0

Find the scalar product for vectors and when

4 units and 5 units for

(a) 45 ( ) 90

a b

a b

b

( ) . cosa a b a b 04 5 cos 45

1220

10 2

( ) . cosb a b a b 04 5 cos90

20 0

0

If a and b are perpendicular then a . b = 0

Component Form of a Scalar Product

1 1

2 2

3 3

If and

a b

a a b b

a b

1 1 2 2 3 3then .a b a b a b a b

1 2

Find . for vectors 3 , 6

4 2

u w u w

. 1 2 3 6 ( 4) 2u w

12

Angle between vectors

We have seen that . cosa b a b

1 1 2 2 3 3and that .a b a b a b a b

1 1 2 2 3 3.Hence cos =

a b a b a ba b

a b a b

NOTE: if 0, 0 and . 0a b a b 0cos 0 90

Hence if . 0, and are perpendiculara b a b

Calculate the angle between vectors

3 +2j 5 and 4 j 3p i k q i k

9 4 25 38p 16 1 9 26q

3 4 2 1 5 3cos

38 26

0.9230 022.7 0 180

Other Vector FactsFor vectors and , . .a b a b b a

PROOF

1 1 2 2 3 3.a b a b a b a b

1 1 2 2 3 3b a b a b a

.b a

For vectors , and , . . .a b c a b c a b a c

PROOF

1 1 2 2 3 31 2 3.a b c a b c a b c a b c

1 1 1 1 2 2 2 2 3 3 3 3a b a c a b a c a b a c

1 1 2 2 3 3 1 1 2 2 3 3a b a b a b a c a c a c

. .a b a c

Calculate . when

3, 3, 4

p r q

p r q

300

300

p

r

q

. . .p r q p r p q

0 03 3cos30 3 4cos60

9 36

2