Unit Vector (Direction)

Statement

A unit vector is a vector that has a length of 1 but keeps the same direction as the original vector. If you have a vector \( \vec{v} = (x, y) \), then its corresponding unit vector is given by:

\[ \hat{u} = \frac{1}{\sqrt{x^2 + y^2}} \begin{pmatrix} x \ y \end{pmatrix} \]

This formula ensures that no matter how large or small the original vector is, the resulting unit vector always has a magnitude of exactly 1.

Why it’s true

• The length (magnitude) of a vector \((x, y)\) is \( \sqrt{x^2 + y^2} \).
• To reduce the length to 1, we divide the whole vector by its magnitude.
• This keeps the same direction because both components are scaled equally.

Recipe (how to use it)

1. Write down the given vector \((x, y)\).
2. Calculate its magnitude \( \sqrt{x^2 + y^2} \).
3. Divide each component by that magnitude.
4. Write the result as the unit vector.

Spotting it

You are asked to find a vector in the same direction but with length 1. Keywords include: unit vector, direction vector, normalised vector.

Common pairings

• Unit vectors are often used in physics problems, especially when defining forces or velocities in a certain direction.
• They also appear in trigonometry and coordinate geometry when working with directions of lines.

Mini examples

1. Given: \((3, 4)\). Find: Unit vector. Answer: \(\left(\tfrac{3}{5}, \tfrac{4}{5}\right)\).
2. Given: \((5, 12)\). Find: Unit vector. Answer: \(\left(\tfrac{5}{13}, \tfrac{12}{13}\right)\).

Pitfalls

• Forgetting to divide both components by the same magnitude.
• Leaving the answer unsimplified when a neat fraction exists.
• Forgetting that the magnitude must always be positive.
• Confusing unit vector with vector of length equal to the magnitude (opposite operation).

Exam strategy

• Always compute the magnitude first; check arithmetic carefully.
• Write your answer as exact fractions where possible, decimals only if requested.
• Check the magnitude of your answer: it should equal 1.

Summary

A unit vector is simply the “scaled down” version of any vector, keeping the same direction but with a fixed length of 1. To find it, divide each component by the vector’s magnitude. This makes it a powerful tool in both pure maths and applied contexts such as physics, mechanics, and computer graphics.