Further Mathematics · Lesson 13

Vectors in 2D

WAECNECO

What the syllabus asks

WAEC: Vectors: definition and representation; column vectors and i, j notation; magnitude and direction; addition, subtraction and scalar multiplication; unit vectors; position vectors; parallel and collinear vectors; ratio theorem; scalar (dot) product and angle between vectors; resolution of vectors; resultant; applications to geometry and relative motion

NECO: Vectors: definition and representation; column vectors and i, j notation; magnitude and direction; addition, subtraction and scalar multiplication; unit vectors; position vectors; parallel and collinear vectors; ratio theorem; scalar (dot) product and angle between vectors; resolution of vectors; resultant; applications to geometry and relative motion

What you will learn

Represent vectors as column vectors, in i, j form and as directed line segments.
Find magnitude, direction and unit vectors.
Add, subtract and scale vectors; find resultants.
Use position vectors, AB = OB − OA, midpoints and the ratio theorem.
Test for parallel vectors and collinear points.
Use the dot product to find angles, test perpendicularity and find projections.
Resolve vectors and solve navigation and geometry problems.

Key points

Vectors have magnitude and direction; |a₁i + a₂j| = √(a₁² + a₂²).
Direction: tan θ = a₂/a₁, adjusted for the quadrant; bearings measured clockwise from north.
Add/subtract by components; ka scales each component.
Unit vector â = a/|a|; vector of length L in direction a is L·â.
AB = b − a; midpoint ½(a + b); ratio m : n gives p = (na + mb)/(m + n).
Parallel: b = ka; collinear: parallel with a common point.
a·b = a₁b₁ + a₂b₂ = |a||b|cos θ; perpendicular if a·b = 0.
Resolve: r cos θ i + r sin θ j; projection of a on b = a·b/|b|.
Relative velocity of A to B = v_A − v_B.

Start here: Scalars, vectors and notation

Some quantities are fully described by a number and a unit: a mass of 50 kg, a time of 3 hours, a temperature of 30 °C. These are scalars. Others also need a direction: "walk 200 m" is incomplete, but "walk 200 m due east" tells you exactly where you end up. These are vectors. Displacement, velocity, acceleration and force are vectors; distance, speed, mass and time are scalars.

Ways to write a vector

Directed line segment: an arrow from A to B, written AB with an arrow over it, or simply AB in bold. Its length is the magnitude; its arrow shows the direction.
Single bold letter: a, b (in handwriting, underline: a).
Column vector: (3, 4) written vertically, meaning 3 units across and 4 units up.
i, j form: 3i + 4j, where i is one unit along the x-axis and j one unit along the y-axis.

All three describe the same object. In this lesson we write vectors in bold and use i, j form most of the time, because WAEC Further Maths questions do.

Two vectors are equal if they have the same magnitude and the same direction, wherever they are drawn. The negative of a: −a has the same length but points the opposite way. AB = −BA.

Worked example 13.1

A taxi in Ibadan drives 3 km east then 4 km north. Write its displacement as a vector and state whether "7 km" describes it.

Taking east as i and north as j, displacement = 3i + 4j km. We combine the two legs as components.
The distance travelled is 7 km (a scalar), but the displacement has magnitude √(3² + 4²) = 5 km. So "7 km" is the distance, not the displacement.

3i + 4j km; magnitude 5 km, not 7 km

⚠️ Common mistake: adding magnitudes of vectors in different directions (3 + 4 = 7). Vectors add by components, not by lengths.

Try it yourself. Classify: speed, velocity, mass, weight, time, force.

Show solution

Scalars: speed, mass, time. Vectors: velocity, weight, force (weight is a force acting downwards).

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Practice questions

1. Find |7i − 24j|.

Show worked answer

25.

2. Find the unit vector in the direction of 8i + 6j.

Show worked answer

0.8i + 0.6j.

3. If a = i + 3j and b = 2i − j, find 3a − 2b.

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−i + 11j.

4. A(−1, 4), B(5, −4): find AB and |AB|.

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6i − 8j; 10.

5. Find m if 2i + mj is perpendicular to 4i − 2j.

Show worked answer

8 − 2m = 0: m = 4.

Key words

Scalar: A quantity with magnitude only, e.g. mass, time, speed.
Vector: A quantity with magnitude and direction, e.g. displacement, velocity, force.
Magnitude: The size (length) of a vector, written |a|.
Unit vector: A vector of magnitude 1, written â.
i, j: Unit vectors along the positive x- and y-axes.
Column vector: A vector written as a column of components.
Position vector: The vector from the origin O to a point.
Resultant: The single vector equal to the sum of several vectors.
Zero (null) vector: A vector of magnitude 0.
Equal vectors: Vectors with the same magnitude and direction.
Negative vector: −a: same magnitude as a, opposite direction.
Parallel vectors: Vectors that are scalar multiples of each other.

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