Angle between two vectors

This online calculator finds the 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

PLANETCALC, Angle between two vectors

Angle between two vectors

First vector

Second vector


Finding the angle between two vectors

We will use the geometric definition of the Dot product 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