Angle Between Two Vectors Formula

Angle Between Two Vectors



The angle between two vectors can be calculated by the arc cosine function of the dot product of the two vectors divided by the product of their respective magnitudes.

Expression Description
The angle between the two vectors
The first vector
The second vector

Usage Notes

Sub Formulas

Dot Product

The dot prodcut of two vectors is calculated by summing together the product of corresponding elements.

Dot Product Formula
Magnitude of Vector

The magnitude of a vector is given by the square root of the sum of its components squared.

Magnitude of Vector Formula


Angle Between Vectors Example
Angle Between Vectors Example

To find the angle between two vectors one can use the formula below:

  1. Substitute the vectors into the formula.

  2. Compute dot product and magnitudes.

  3. Evaluate multiplication, exponent and square-root expressions. Since the numerator goes to zero, the whole expression goes to zero.

  4. Compute the arc-cosine.

  5. The angle is radians or .