Online calculator: Angle between two vectors

This calculator finds the angle between two vectors given their coordinates. The formula and the explanation can be found below the calculator.

Angle

We will use the geometric definition of the 3D Vector Dot Product Calculator to produce the formula for finding the angle.

Geometrically the dot product is defined as

$\mathbf {a} \cdot \mathbf {b} =\|\mathbf {a} \|\ \|\mathbf {b} \|\cos(\theta )$

thus, we can find the angle as

$\theta=arccosine\left(\frac{\mathbf {a} \cdot \mathbf {b}}{||\mathbf {a}|| \cdot ||\mathbf {b}||}\right)$

To find the dot product from vector coordinates, we can use its algebraic definition.

Thus, for two vectors, $a=[x_1, y_1, z_1]$ and $b=[x_2, y_2, z_2]$ , formula can be written as

$\theta=arccosine\left(\frac{x_1x_2+y_1y_2+z_1z_2}{\sqrt{x_1^2 + y_1^2 + z_1^2} \cdot \sqrt{x_2^2+y_2^2+z_2^2}}\right)$

This is the formula used by the calculator.

PLANETCALC Online calculators