How Do You Find a Unit Vector?

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This is IB Mathematics: Analysis and Approaches HL content from AHL 3.12: Vector definitions. It may be tested on Papers 1, 2, or 3. The exact example below is suitable for Paper 1, where calculators are not permitted.

The Method

For a vector

first calculate its magnitude using Pythagoras in three dimensions:

Then divide every component by this magnitude:

Each component must be divided by the same positive scalar. Consequently, the component ratios remain unchanged, so the new vector is parallel to and points in the same direction as the original vector.

Worked Example

Find the unit vector in the direction of

First find its magnitude:

Therefore,

Verify the result:

A common misconception is to divide by the sum of the components. Instead, divide by the square root of the sum of their squares. The zero vector has no unit vector because its magnitude is $0$, making division undefined.

Direction also matters. The unit vector from point A to point B is the negative of the unit vector from B to A. This happens because reversing the displacement reverses every component while leaving the magnitude unchanged. Therefore, always read the requested direction before subtracting position vectors. Multiplying a vector by a positive scalar preserves its direction, whereas multiplying it by a negative scalar reverses its direction. In both cases, normalization still produces a vector of magnitude one.

Exam Technique

Show both stages: calculate $|\mathbf{v}|$, then write $\mathbf{v}/|\mathbf{v}|$. If points $A$ and $B$ are given, first form $\overrightarrow{AB}=\mathbf{b}-\mathbf{a}$. Reversing the subtraction produces a unit vector in the opposite direction. Keep an exact answer on Paper 1 unless the question requests an approximation.