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

A vector is a quantity that has both magnitude and direction, and the magnitude (or modulus) of a vector a is written |a|.

Behavioural objectives

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

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

What is a vector?

A vector is a quantity with both size (magnitude) and direction.

• Scalar: a quantity with magnitude only, such as mass, time, speed and temperature.
• Vector: a quantity with magnitude and direction, such as displacement, velocity and force.

How is a vector written in terms of i and j?

A vector in two dimensions is written as xi + yj, where i and j are unit vectors along the x-axis and y-axis.

• i: the unit vector in the direction of the positive x-axis.
• j: the unit vector in the direction of the positive y-axis.
• x: the number of units in the i direction (horizontal part).
• y: the number of units in the j direction (vertical part).

Example: 3i + 4j means 3 units along the x-axis and 4 units along the y-axis.

What is the magnitude of a vector?

The magnitude of a vector, also called its modulus, is its length, and it is written |a|.

For r = xi + yj, the modulus is |r| = √(x² + y²).

Steps for finding the modulus

1. Write the vector in the form xi + yj.
2. Square the number with i and the number with j.
3. Add the two squares.
4. Take the square root of the sum.

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

Note: a negative sign disappears after squaring, so a modulus is never negative.

What are the special types of vector?

The special vectors are the zero vector, the unit vector and the negative vector, and there is also a rule for equal vectors.

• Zero vector: a vector with zero magnitude, written 0i + 0j.
• Unit vector: a vector with magnitude 1, written â, where a = |a|â and â = a/|a|.
• Negative vector: the vector -a, with the same magnitude as a but opposite direction.
• Equal vectors: two vectors with the same magnitude and direction.

Equal vectors in terms of i and j

• xi + yj = pi + qj only when x = p and y = q.
• The i parts must match, and the j parts must match.

Example: finding a unit vector

Find the unit vector in the direction of a = 3i + 4j.

1. |a| = √(3² + 4²) = 5.
2. â = a/|a| = (3i + 4j)/5.
3. â = (3/5)i + (4/5)j.

Example: finding a negative vector

• If a = 3i – 2j, then -a = -3i + 2j.
• Every sign in the vector changes.

How do you carry out arithmetic operations on vectors?

You add, subtract and multiply vectors by working on the i parts and the j parts separately.

Rules

• Addition: add the i parts together and add the j parts together.
• Subtraction: subtract the i parts and subtract the j parts.
• Multiplication by a number: multiply both the i part and the j part by that number.

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

What mistakes should you avoid?

• Mixing i and j: never add an i part to a j part.
• Sign errors in subtraction: bracket the second vector, so -(6i + 10j) becomes -6i – 10j.
• Forgetting the square root: the modulus is √(x² + y²), not x² + y².
• Writing a negative modulus: magnitude is always zero or positive.
• Wrong order in CD: CD = OD – OC, not OC – OD.

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.
5. Write down the negative vector of 4i – 7j.

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.