Magnitude and Direction of a Vector - Calculator

An online calculator to calculate the magnitude and direction of a vector from its components.

Let \( \vec{v} \) be a vector given in component form by

\[ \vec{v} = \langle v_1 , v_2 \rangle \]

The magnitude \( \| \vec{v} \| \) of vector \( \vec{v} \) is given by

\[ \| \vec{v} \| = \sqrt{v_1^2 + v_2^2} \]

The direction of vector \( \vec{v} \) is the angle \( \theta \) in standard position such that

\[ \tan(\theta) = \frac{v_2}{v_1}, \quad 0 \le \theta < 2\pi \]

Use of the Calculator

Enter the components \( v_1 \) and \( v_2 \), then compute the magnitude \( \| \vec{v} \| \) and direction \( \theta \) (in degrees).

\( v_1 = \) , \( v_2 = \)

Decimal Places =
Magnitude: \[ \| \vec{v} \| = \]
Direction: \[ \theta = \] \( ^\circ \)

Practice Questions

  1. Find the direction of \( \vec{u} = \langle -2 , 3 \rangle \) and \( \vec{v} = \langle -4 , 6 \rangle \). Why are they equal?
  2. Find the direction of \( \vec{u} = \langle 2 , 5 \rangle \) and \( \vec{v} = \langle -2 , -5 \rangle \). Why is the difference \( 180^\circ \)?
  3. Find the direction of \( \vec{u} = \langle 2 , 1 \rangle \) and \( \vec{v} = \langle 1 , 2 \rangle \). Why do they sum to \( 90^\circ \)?