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Level: JHS (Basic 7 – 9) | Subject: Mathematics | Topic: Vectors in Two Dimensions | Curriculum: Ghana NaCCA / GES

Vectors are quantities that have both magnitude (size) and direction. They appear in Physics as well as Mathematics and are one of the topics that JHS students often find conceptually challenging. This lesson builds the foundation clearly, from basics through to vector addition, subtraction, and simple applications.

Scalars and Vectors

Before defining vectors, it helps to understand the difference between scalars and vectors:

The key distinction: “5 km” is a scalar (distance). “5 km North” is a vector (displacement). Direction makes the difference.

Representing Vectors

Notation

• A vector from point A to point B is written as AB (with an arrow above: →AB) or in bold as a.
• The magnitude (length) of vector AB is written |AB| or simply AB without the arrow.
• Vectors can be represented as column vectors: the top number shows horizontal movement (positive = right, negative = left), the bottom number shows vertical movement (positive = up, negative = down).

Column Vector Notation

The vector ( 3 ) means: move 3 units to the right and 2 units up.

The vector ( -4 ) means: move 4 units to the left and 3 units down.

(Note: column vectors are written as a fraction-style pair with the x-component on top and y-component below.)

Written out:

• a = (3, 2): 3 right, 2 up
• b = (-4, -3): 4 left, 3 down
• c = (0, 5): no horizontal movement, 5 up

Types of Vectors

Vector Addition

When vectors are added, you place them end to end (tip to tail). The sum is the resultant vector from the start of the first to the end of the last.

Adding Column Vectors

Simply add the corresponding components:

If a = (3, 2) and b = (1, -4), then:

a + b = (3+1, 2+(-4)) = (4, -2)

More examples:

• (5, 3) + (2, 4) = (7, 7)
• (-3, 6) + (5, -2) = (2, 4)
• (4, -1) + (-4, 1) = (0, 0) = zero vector

The Triangle Law

If vectors a and b are drawn tip to tail, the resultant (a + b) is the vector from the tail of a to the tip of b. This is the triangle law of vector addition.

Vector Subtraction

Subtracting a vector is the same as adding its negative: a – b = a + (-b).

If a = (6, 4) and b = (2, -1), then:

a – b = (6-2, 4-(-1)) = (4, 5)

Scalar Multiplication of Vectors

Multiplying a vector by a scalar (number) multiplies each component:

• 3 × (2, -1) = (6, -3)
• 2 × (4, 5) = (8, 10)
• -1 × (3, 7) = (-3, -7) [this reverses the direction]
• ½ × (8, -4) = (4, -2)

Magnitude of a Vector

The magnitude (length) of a vector (x, y) is found using Pythagoras’ theorem:

|v| = √(x² + y²)

Examples:

• |(3, 4)| = √(3² + 4²) = √(9 + 16) = √25 = 5
• |(5, 12)| = √(25 + 144) = √169 = 13
• |(6, 8)| = √(36 + 64) = √100 = 10

These are Pythagorean triples (3-4-5, 5-12-13, 6-8-10), which appear frequently in BECE exam questions on vectors.

Position Vectors

A position vector describes the position of a point relative to the origin (0, 0). If point A is at coordinates (3, 5), the position vector of A is OA = (3, 5).

If A has position vector a and B has position vector b, then vector AB = b – a.

Practice Questions: Vectors

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