Magnitude and Resultant Vectors: GCSE Maths Guide

The magnitude of a vector is its length. For a column vector a = (x, y), the magnitude is |a| = √(x² + y²) by Pythagoras. The resultant of two vectors is their combined effect: add the column vectors component by component. Scalar multiplication stretches or shrinks a vector without changing its direction.

How do you write and read column vectors?

A column vector is written as a vertical pair of numbers: a = (p, q), where p is the horizontal component (positive = right) and q is the vertical component (positive = up).

At GCSE, vectors are also named with bold letters (a, b) or underlined letters (a, b) in handwritten work.

How do you calculate the magnitude of a vector?

Apply Pythagoras to the two components. The horizontal and vertical components are the two shorter sides; the magnitude is the hypotenuse.

|a| = √(p² + q²)

Worked examples:

1. a = (3, 4): |a| = √(9 + 16) = √25 = 5
2. b = (5, 12): |b| = √(25 + 144) = √169 = 13
3. c = (−4, 3): |c| = √(16 + 9) = √25 = 5 (magnitude is always positive)
4. d = (2, 7): |d| = √(4 + 49) = √53 ≈ 7.28 (to 3 s.f.)

Note: the negative sign on a component does not affect the magnitude — it is squared away.

How do you find the resultant of two vectors?

The resultant vector is found by adding the two column vectors — add the top components together and the bottom components together.

a + b = (p₁ + p₂, q₁ + q₂)

Worked example 1:

a = (3, 5) and b = (−1, 4).

a + b = (3 + (−1), 5 + 4) = (2, 9)

|a + b| = √(4 + 81) = √85 ≈ 9.22

Worked example 2 — subtraction:

a − b = (p₁ − p₂, q₁ − q₂)

a − b = (3 − (−1), 5 − 4) = (4, 1)

Subtracting b is the same as adding −b, which reverses the direction of b.

What is scalar multiplication?

A scalar is an ordinary number (not a vector). Multiplying a vector by a scalar k scales every component:

ka = (kp, kq)

The direction is unchanged (same or reversed if k < 0); the magnitude is scaled by |k|.

Worked examples:

Note that |ka| = |k| × |a|. For the first example: |3a| = 3 × √(4 + 1) = 3√5 ✓.

How do you find a resultant in a geometry problem?

GCSE vector geometry problems describe a path using vector labels and ask for the resultant of a journey.

Worked example 3:

OA = a = (2, 3) and OB = b = (5, 1). Find the vector AB and its magnitude.

AB = AO + OB = −a + b = −(2, 3) + (5, 1) = (3, −2)

|AB| = √(9 + 4) = √13 ≈ 3.61

The key step is identifying direction: AB goes FROM A TO B, so it starts with AO (the reverse of OA) then adds OB.

Worked example 4:

M is the midpoint of AB where A = (1, 4) and B = (7, 2). Find OM.

OM = OA + ½AB = (1, 4) + ½(6, −2) = (1, 4) + (3, −1) = (4, 3)

This is equivalent to the midpoint formula: ((1+7)/2, (4+2)/2) = (4, 3) ✓.

Frequently asked questions

What is the difference between a vector and a scalar?

A vector has both magnitude (size) and direction; a scalar has magnitude only. Distance is a scalar (e.g. 10 m), but displacement is a vector (e.g. 10 m north). At GCSE, speed is scalar and velocity is vector — but in the maths paper, focus on column vectors and their arithmetic.

Why is the magnitude always positive?

Because it represents a physical length, which cannot be negative. Even if both components are negative (e.g. (−3, −4)), the magnitude is √(9 + 16) = 5. The direction of a vector is captured by the signs of the components, not by its magnitude.

How do I find a unit vector at GCSE?

A unit vector has magnitude 1. To find the unit vector in the direction of a, divide a by its magnitude: â = a/|a|. This is not commonly asked at GCSE but may appear as a stretch question. For a = (3, 4), â = (3/5, 4/5) and |(3/5, 4/5)| = √(9/25 + 16/25) = √1 = 1 ✓.

Can vectors have three components?

In 3D problems, vectors have three components (x, y, z). Three-dimensional vectors are not in the GCSE specification, though 3D Pythagoras and trigonometry are. For GCSE, all column vectors are two-dimensional.

Work through vector problems with Professor Pi — visit aitutors.me.