Practice Vector Magnitude and Direction in Math
Use these practice problems to test your method after reviewing the concept explanation and worked examples.
Quick Recap
The magnitude (or length) of a vector \mathbf{v} = \langle v_1, v_2 \rangle is \|\mathbf{v}\| = \sqrt{v_1^2 + v_2^2}, calculated using the Pythagorean theorem. A unit vector has magnitude 1 and indicates direction only. The unit vector in the direction of \mathbf{v} is \hat{\mathbf{v}} = \frac{\mathbf{v}}{\|\mathbf{v}\|}.
Magnitude is how long the arrow is—like measuring the length of a stick. Direction is which way it points. A unit vector is a 'pure direction' with length 1, like a compass needle. To get the unit vector, shrink or stretch the vector until its length is exactly 1 while keeping it pointed the same way.
Example 1
easyFind the magnitude of \mathbf{v} = \langle 3, 4 \rangle.
Example 2
mediumFind the unit vector in the direction of \mathbf{v} = \langle 1, 2, 2 \rangle.
Example 3
easyFind \|\langle -5, 12 \rangle\|.
Example 4
hardFind the direction angle \theta of \mathbf{v} = \langle -1, \sqrt{3} \rangle measured from the positive x-axis.