All subjects · Mathematics · WME01 (M1)

Vectors: magnitude, direction and resultants

Displacement, velocity, acceleration and force are vectors: they have a size and a direction. M1 writes them with the unit vectors i and j.

Magnitude and direction

For a = xi + yj:

magnitude |a| = sqrt(x² + y²) (Pythagoras)
the angle with i is found from tan θ = y/x. Always sketch the vector to see which quadrant it is in.

If i is east and j is north, a bearing is measured clockwise from north (j).

Resultants

The resultant of several vectors is their sum. Add the i parts and add the j parts: (2i + 3j) + (4i - 11j) = 6i - 8j.

A vector is parallel to i when its j part is zero; parallel to i + j when its two parts are equal.

Unit vectors and scaling

A unit vector has magnitude 1. To make a vector of magnitude k in the direction of a, use k × a/|a|. For example, a force of 15 N in the direction 3i + 4j is 9i + 12j N.

Worked example

Find the magnitude of the resultant of 2i + 3j and 4i - 11j, and the angle it makes with i.

Resultant = 6i - 8j.
Magnitude = sqrt(36 + 64) = 10.
tan θ = 8/6, so θ = 53.1° below i.

Answer: 10, at 53.1° below the direction of i

Key idea: Add components to find a resultant; magnitude by Pythagoras; direction by tan and a sketch.

Check yourself

Find the magnitude of 5i - 12j.
Show answer
13. sqrt(25 + 144) = sqrt(169) = 13.
Find the angle between 3i + 4j and the unit vector i, in degrees to 1 decimal place.
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53.1 °. tan θ = 4/3, so θ = 53.1°.
Find the magnitude of the resultant of 7i + 2j and -i + 6j.
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10. Resultant = 6i + 8j, magnitude sqrt(36 + 64) = 10.
With i east and j north, find the bearing of the vector 3i - 3j.
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135 °. It points south-east, so its bearing is 135°.
The resultant of the forces (3i + pj) N and (4i + (p - 3)j) N is parallel to i + j. Find p.
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5. Resultant = 7i + (2p - 3)j. Parallel to i + j means equal components: 2p - 3 = 7, so p = 5.
A force has magnitude 26 N and acts in the direction of 5i + 12j. Find the coefficient of j in this force.
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24 N. |5i + 12j| = 13, so the force is (26/13)(5i + 12j) = 10i + 24j. The j coefficient is 24.
Which vector is parallel to i?
- 4i + 4j
- 4j
- i - j
- -4i
Show answer
-4i. A vector parallel to i has no j part.

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Last updated 7 November 2026. Spotted a mistake? Tell us.