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

The magnitude (or modulus) of a vector a is written |a|. For a vector written in the form xi + yj, the modulus is the square root of x² + y². Two vectors are equal when they have the same magnitude and direction. This note also covers adding, subtracting and multiplying vectors by numbers.

Behavioural objectives

By the end of the lesson, students should be able to:

• state what is meant by the magnitude (modulus) of a vector;
• describe the zero vector, the unit vector and the negative vector, and say when two vectors are equal;
• find the modulus of a vector written in the form xi + yj;
• add, subtract and multiply vectors written in terms of i and j;
• find a vector such as CD from given vectors OC and OD.

What is the magnitude of a vector?

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

Types of vector

1. Zero vector: The zero vector is a vector with zero magnitude.
2. Unit vector: The unit vector is the vector represented by â and is such that a = |a|â.
3. Negative vector: The negative vector of a is written as -a.
4. Equality of vectors: Two vectors are equal when they have the same magnitude and direction.

Example: finding the modulus

Find the modulus of each of the following vectors.

1. 3i + 4j
2. -2i – 5j

Solution

1. Let r = 3i + 4j, then
   |r| = √(3² + 4²) = √25 = 5
2. Let r = -2i – 5j, then
   |r| = √((-2)² + (-5)²) = √(4 + 25) = √29

How do you carry out 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(2i – 3j) = 6i – 9j
   2q = 2(3i + 5j) = 6i + 10j
   Therefore 3p – 2q = (6i – 9j) – (6i + 10j)
   = -19j

Example 2

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

Solution

CD = CO + OD = OD – OC
= (2a + 3b) – (a – b)
= 2a + 3b – a + b
= a + 4b
= (2i + 3j) + 4(3i – 2j)
= 14i – 5j

Evaluation

1. What is meant by the modulus of a vector?
2. When are two vectors equal?
3. Find the modulus of 6i + 8j.
4. If p = 2i – 3j and q = 3i + 5j, find 2p + q.

Assignment

1. Find the modulus of each vector: (a) -5i + 12j (b) 2i – 4j.
2. If a = i + 2j and b = 3i – j, find 2a + 3b.
3. If a = i + 2j and b = 3i – j, find 3a – 2b.