Unit vectors

A unit vector is a vector with a magnitude (length) of exactly 1. It shows a direction and nothing else. To find the unit vector in the direction of any vector a, divide a by its own magnitude: â = a ÷ |a|. For example, the vector (3, 4) has magnitude 5, so its unit vector is (0.6, 0.8). The special unit vectors i, j and k point along the x, y and z axes.

What is a unit vector?

Every vector has a size and a direction. A unit vector keeps the direction but has its size set to 1. It is written with a hat: â is read "a hat" and means the unit vector in the direction of a. Turning a vector into its unit vector is called normalising it (normalizing in the US).

The unit vector formula

For a 2D vector (x, y), the magnitude comes from Pythagoras, and in 3D you add the third component:

How to find a unit vector

1. Find the magnitude |a| using Pythagoras.
2. Divide every component of a by |a|.
3. Check: the magnitude of your answer should be exactly 1.

Example 1: a 2D vector

Find the unit vector in the direction of a = (3, 4).

Check: 0.6² + 0.8² = 0.36 + 0.64 = 1. It works.

Example 2: i and j notation

Find the unit vector in the direction of b = 5i − 12j.

Example 3: a 3D vector

Find the unit vector in the direction of c = 2i − j + 2k.

The Vectors calculator finds magnitudes and unit vectors in 2D and 3D and shows the working.

The unit vectors i, j and k

The most important unit vectors point along the axes: i along the x-axis, j along the y-axis and k along the z-axis. Any vector can be written as a sum of them:

So 2i − j + 2k is the same vector as (2, −1, 2). In 2D only i and j are needed. AQA's A-level Maths specification lists â and i, j, k in its notation for vectors, which cover two and three dimensions.

Using unit vectors

Example 4: the unit vector from one point to another

Find the unit vector in the direction from A(1, 2) to B(4, 6).

First find the vector AB by subtracting: AB = (4 − 1, 6 − 2) = (3, 4). Its magnitude is 5, so the unit vector is (0.6, 0.8).

Example 5: a vector of a given length

Find a vector of magnitude 10 in the direction of (1, 2, 2).

The magnitude of (1, 2, 2) is √(1 + 4 + 4) = 3, so the unit vector is (1/3, 2/3, 2/3). Multiply by 10:

This is the most common reason to find a unit vector: it lets you build a vector of any length in a chosen direction.

Example 6: from an angle

A unit vector at angle θ to the positive x-axis is (cos θ, sin θ), because its magnitude is √(cos²θ + sin²θ) = √1 = 1. At 30°, it is (√3/2, 1/2).

Example 7: a velocity

A boat moves at 15 m/s in the direction of 3i + 4j. Its velocity vector is the speed times the unit vector: 15 × (0.6i + 0.8j) = 9i + 12j m/s.

Example 8: the angle a vector makes with an axis

The components of a unit vector are the cosines of the angles it makes with the axes. For ĉ = (2/3, −1/3, 2/3) from example 3, the angle with the x-axis is cos⁻¹(2/3) = 48.2° and the angle with the y-axis is cos⁻¹(−1/3) = 109.5°. In 2D, the unit vector (0.6, 0.8) makes an angle of cos⁻¹(0.6) = 53.1° with the x-axis. This is a quick way to find a vector's direction once you have its unit vector.

Check your understanding.

Common mistakes

• Dividing by the sum of the components instead of the magnitude. For (3, 4), divide by 5, not 7.
• Forgetting to square negative components. (−12)² = 144, which is positive.
• Dividing only one component. Every component must be divided by |a|.
• Missing the third component of a 3D vector when finding the magnitude.
• Not checking. Squaring and adding your answer's components must give exactly 1.

Practise with a short timed paper

Vectors come up in GCSE Maths Higher, A-level Maths and university linear algebra. The daily GCSE Maths Higher and Linear Algebra papers are short timed papers of 8 questions, with a new one every day.

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