Vectors in Two Dimensions – SS1 Further Mathematics Lesson Note (Second Term)

Magnitude of a vector

The magnitude of a vector, sometimes called the modulus of the vector is represented by |a|.

1. Zero Vector: The zero vector is a vector with zero magnitude.

2. Unit Vector: The unit vector is the vector represented by a and is such that a = |a| a

3. Negative Vector: The negative vector of a is written as – a 

4. Equality of vector: Two vectors are equal when they have the same magnitude and direction.

Example:

Find the modulus of each of the following vectors

1. 3i + 4j

2. -2i – 5j

Solution

1. Let r = 3i + 4j ; then 

|r| = ✓(3^2) + (4^2) = ✓25 = 5

2. Let r = -2i – 5j ; then

|r| = ✓(-2^) – (5^2) = ✓4+25 = ✓29

Arithmetic operations on vectors

Example 1:

If p = 2i –  3j; q =  3i + 5j and r = i + j; Find the values of:

1. 2p + q + 3r
2. 3p – 2q 

Solution

1. 2p = 2(2i – 3j ) = 4i – 6j 

3r = 3( i + j ) = 3i + 3j

Therefore; 2p + q + 3r = (4i – 6j) + (3i + 5j) + (3i + 3j)

= 10i + 2j

2. 3p = 3(3i – 3j) = 9i – 9j

2q = 2(3i + 5j) =  6i + 10j

Therefore 3p – 2q = (9i – 9j) – (6i + 10j)      =3i – 19j

Example 2:

Given that OC = a – b and = 2a + 3b, where a = 2i + 3j and b = 3i – 2j, find CD

CD = CO + OD  = OD – OC

       = (2a + 3b) – (a – b)

      = 2a + 3b – a + b = a + 4b

      = (2i + 3j) + 4(3i – 2j)   = 14i – 5j