Practice Maths

Introduction to Vectors

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Key Terms

Vector
A quantity with both magnitude AND direction (e.g. displacement, velocity, force). Written in bold or with an arrow.
Scalar
A quantity with magnitude only, no direction (e.g. distance, speed, mass).
Magnitude |a|
The length (size) of the vector; for a = (x, y), |a| = √(x² + y²).
Position vector
The vector OP from the origin O to the point P; uniquely locates P.
Unit vector
A vector with magnitude 1; â = a/|a|; the standard unit vectors are i = (1,0) and j = (0,1).
Column vector notation
A vector can be written as a column: (x, y) or as xi + yj.

Core Vector Definitions

A vector has both magnitude (size) and direction. A scalar has magnitude only.

Notation: v (bold), v (underline), or component form xi + yj

Standard unit vectors:   i = (1, 0) points in the positive x-direction;   j = (0, 1) points in the positive y-direction.

Position vector of point P(x, y):   OP = xi + yj

Magnitude:   |v| = √(x2 + y2)   where   v = xi + yj

Unit vector:   v̂ = v / |v|   (always has magnitude 1)

Equal vectors: same magnitude AND same direction (position in the plane does not matter).

Negative vector: −a has the same magnitude as a but points in the opposite direction.

Zero vector: 0 = 0i + 0j, magnitude 0, no defined direction.

Worked Example 1 — Finding Magnitude

Find the magnitude of v = 3i + 4j.

|v| = √(32 + 42) = √(9 + 16) = √25 = 5

Worked Example 2 — Finding a Unit Vector

Find the unit vector in the direction of a = 6i − 8j.

Step 1: Find the magnitude: |a| = √(62 + 82) = √(36 + 64) = √100 = 10

Step 2: Divide by magnitude: â = (6i − 8j) / 10 = (3/5)i − (4/5)j

Check: |(3/5)i − (4/5)j| = √(9/25 + 16/25) = √(25/25) = 1 ✓

Worked Example 3 — Position Vector

State the position vector of point P(−2, 5).

OP = −2i + 5j

This vector points from the origin O(0, 0) directly to P(−2, 5).

Hot Tip — Verify Unit Vectors: After finding v̂, always verify by computing its magnitude. You should get exactly 1. If not, recheck your arithmetic. This quick check catches most errors before they cascade.
Vector typeDefinitionExample
Position vectorFrom origin O to point POP = 3i + 2j for P(3, 2)
Unit vector|v| = 1i, j, (3/5)i + (4/5)j
Zero vectorMagnitude 00 = 0i + 0j
Free vectorEqual if same mag. & dir.Any translate of a

Why Vectors?

Many real-world quantities require both a size and a direction to describe them completely. A car travelling at 60 km/h is described by a scalar (speed). But if you need to navigate, you need to know which direction the car is going — that is a velocity, a vector. Force, displacement, acceleration, electric fields, and gravitational fields are all vectors.

Vectors allow us to describe these quantities algebraically and geometrically, and to combine them using mathematical rules. This makes them indispensable in physics, engineering, computer graphics, and navigation.

Vectors as Directed Line Segments

Geometrically, a vector is drawn as an arrow. The tail is where the arrow starts; the head (arrowhead) is where it ends. The length of the arrow represents the magnitude; the direction the arrow points is the direction.

Crucially, two arrows represent the same vector if and only if they have the same length and point in the same direction — regardless of where they are placed in the plane. This is the concept of a free vector. Translating an arrow anywhere in the plane does not change the vector it represents.

Component Form and the Standard Unit Vectors

Any vector in the plane can be broken into two perpendicular components: one in the horizontal direction and one in the vertical direction. We define:

• i = the unit vector in the positive x-direction = (1, 0)

• j = the unit vector in the positive y-direction = (0, 1)

Every vector v = (x, y) can then be written as v = xi + yj. The numbers x and y are called the components of the vector. They tell us how far to move horizontally and vertically to get from the tail to the head.

Position Vectors

A position vector is a special vector whose tail is fixed at the origin O. The position vector of point P(a, b) is written OP = ai + bj. Position vectors establish a one-to-one correspondence between points in the plane and vectors from the origin — this is extremely useful in coordinate geometry and proofs.

Magnitude: The Length of a Vector

The magnitude of v = xi + yj is its length as an arrow. By Pythagoras’ theorem (since the horizontal and vertical components are perpendicular), |v| = √(x2 + y2). This is the Euclidean distance from origin to the point (x, y).

Key properties: |v| ≥ 0, and |v| = 0 if and only if v = 0.

Unit Vectors and Normalisation

A unit vector has magnitude exactly 1. The process of finding a unit vector in a given direction is called normalisation: divide the vector by its magnitude.

Given any non-zero vector v, the unit vector in its direction is v̂ = v / |v|. This preserves direction but scales the length to exactly 1.

Unit vectors are useful when you care about direction but not magnitude — for example, to describe the direction of travel of a particle, or to find a vector of specific magnitude in a given direction (multiply the unit vector by the desired magnitude).

Equal and Negative Vectors

Two vectors are equal if and only if they have the same components: a = a1i + a2j equals b = b1i + b2j iff a1 = b1 AND a2 = b2.

The negative of vector a = a1i + a2j is −a = −a1i − a2j. It has the same magnitude but opposite direction. Note that |−a| = |a|.

Mastery Practice

  1. Fluency

    Q1 — Magnitudes

    Find the magnitude of each vector:   (a) 5i + 12j    (b) −3i + 4j    (c) 8i − 6j

  2. Fluency

    Q2 — Position Vectors

    State the position vector of each point:   (a) A(3, 7)    (b) B(−2, 5)    (c) C(0, −4)

  3. Fluency

    Q3 — Unit Vectors

    Find the unit vector in the direction of:   (a) a = 4i + 3j    (b) b = −5i + 12j

  4. Fluency

    Q4 — Negative Vectors

    Given a = 2i − j, write the negative vector −a and verify that |−a| = |a|.

  5. Understanding

    Q5 — Finding a Point from a Vector

    Vector PQ = 5i + 2j. The starting point is P(1, 3). Find the coordinates of Q.

  6. Understanding

    Q6 — Vector Between Two Points

    Given A(2, 5) and B(6, 2), find the vector AB and its magnitude.

  7. Understanding

    Q7 — Parallel Unit Vectors

    Find all unit vectors parallel to v = 3i − 4j. (There are two answers.)

  8. Understanding

    Q8 — Vector with Given Magnitude

    A force of 13 N acts in the direction of 5i + 12j. Write the force as a vector.

  9. Problem Solving

    Q9 — Angle of a Unit Vector

    Verify that u = (1/√2)i + (1/√2)j is a unit vector. What angle does it make with the positive x-axis?

  10. Problem Solving

    Q10 — Equal Vectors and Unknown Components

    Two vectors are equal: a = (2t − 1)i + 3j and b = 5i + (t + 5)j. Find t and hence write both vectors explicitly.

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