Magnitude of a vector

Maths · Transformations and vectors. The idea in plain words, the rules to remember, the mistake to avoid and a worked example, written and checked by Mega Lecture.

Test this topic free All Maths topics

The idea

The magnitude of a vector is its length. For a column vector, the two parts are the sides of a right-angled triangle, so the magnitude comes from Pythagoras’ theorem. It is never negative.

The position vector of a point A is the vector from the origin O to A, usually written a. To go from A to B, travel back to O and then out to B, so vector AB = b − a.

In geometry problems, write any vector as a route along lines you already know, such as AB = AO + OB. Going against an arrow changes the sign. Two vectors are parallel when one is a number times the other, and that number compares their lengths.

Rules to remember

Magnitude of the vector (x, y) = √(x² + y²)
Vector AB = position vector of B − position vector of A (AB = b − a)
Vector BA = −(vector AB)
Midpoint M of AB: OM = (1/2)(a + b)
P on AB with AP : PB = m : n: OP = a + (m ÷ (m + n)) × (b − a)
Parallel: if vector PQ = k × vector RS, then PQ is parallel to RS and k times as long
Three points A, B, C lie on one straight line if AB and BC are parallel, because they share the point B

Common mistake

For a point that divides AB in the ratio 1 : 3, students use the fraction 1/3. The line is split into 1 + 3 = 4 parts, so the fraction is 1/4.

Worked example

In triangle OAB, vector OA = a and vector OB = b. M is the midpoint of OA. N is on OB with ON : NB = 3 : 1. Express vector MN in terms of a and b.

OM = (1/2)a, so MO = −(1/2)a.
ON is 3 parts out of 3 + 1 = 4, so ON = (3/4)b.
MN = MO + ON = −(1/2)a + (3/4)b.

Answer: MN = (3/4)b − (1/2)a

Ask about this

Type what you are stuck on. The helper finds the explanation, worked examples and the notes to read, free and instant.

Read the full notes

Watch: Vectors

Keep going

← Vectors in two dimensions Introduction to probability →Take a Maths test