VectorsThe modulus of a vector is its magnitude.

Unit vectors are vectors with unit magnitude.

The modulus of a is written as .a

The modulus of is written as .AB��������������

AB��������������

is the unit vector in the direction of a.a

i, j and k are unit vectors parallel to the axes

Ox, Oy and Oz

Position vector and its magnitude

a

OP ai bj ck b

c

�������������� 2 2 2OP a b c ��������������

AB OB OA �������������� ����������������������������

Displacement vector

Vectors

The points A, B and C have coordinates (4, -2), (3, 3) and (-2, 1) respectively and O is the origin.

Find in terms of i and j , the vectors

Example

( i ) OA��������������

( ii ) AB��������������

( iii ) CB��������������

( iv ) AC��������������

( v ) CA��������������

4 2( i ) i j 5( ii ) i j 5 2( iii ) i j

6 3( iv ) i j 6 3( v ) i j

Parallel vectorsa a

b s b

c c

sa

sb

sc

is parallel to a

b

c

and is s times its length.

Unit vectorsMagnitude of a vector If a xi yj zk

2 2 2Then magnitude of a a x y z

1a

unit vector a ( xi yj zk )

Example

Find the magnitude of each these vectors

6 8( i ) i j 12 5( ii ) i j 2 3 5( iii ) i j k

2 26 8 10( i ) 2 212 5 13( ii ) 2 2 22 3 5 38( iii ) Unit vectors in these directions are

110 6 8( i ) ( i j ) 1

13 12 5( ii ) ( i j ) 1382 3 5( iii ) ( i j k )

Examplep = 4i + j q = 6i -

5jr = 3i + 4jFind numbers m and n such that mp + nq = r.

i component: 4m + 6n = 3

[1]

j component: m - 5n = 4 [2]

[1] – 4[2]: 26n = - 13

n = - ½

m = 1.5

Example

Find a vector of magnitude 30 in the direction of 6i -8j

2 26 8 10

3010 6 8 18 24( i j ) i k

2 2

5 1

3 1

If OA and OB

����������������������������

Find

(i) the vector AB

(ii) AB

(iii) Position vector of the mid-point of AB

2 2 0

1 5 6

1 3 2

AB

�������������� 2 26 2 40 2 10AB ��������������

2 2 21 1

1 5 22 2

1 3 2

Mid point

(i) (ii)

(iii)

Example

ExampleGiven that v = i +2j – 6k, find such that v= 7

22 2 2a b c v 2 2 22 6 49

3

Examplev1 = 3i + 2j – k, r2 = 2i – k. Find the unit vector in the direction of 2v1 – v2.

1 2

3 2 4

2 2 2 0 4

1 1 1

v v

1334 4unit vector ( i j k )

The length of 2v1 – v2 =33