Vector A
Vector B

Angle Between Two Vectors Calculator

The angle between two vectors is referred to as the shortest angle, having a single starting point and the vectors are directed in different coordinates. These vectors are co-directional to each other. The exact definition of the angle between two vectors is equal to the dot product of the vectors divided by the product of the intensity or magnitude of the vector.

Unit of angle:

The unit of the angle is the degree (°).

Angle Between Two Vectors Calculator Formula

The angle between two vectors can be calculated by the formula:

\[\cos \theta = \frac{{A.B}}{{|A||B|}}\]

Here,

  1. “θ” is the angle between the vectors
  2. “A” is the 1st vector
  3. “B” is the 2nd vector
  4. |A| is the magnitude of 1st angle
  5. |B| is the magnitude of 2nd angle
  6. “.” Is representing the dot product of the vectors

Example

To get an idea of how to calculate the angle between the vectors, we have illustrated a detailed example in the following lines.

The vector A has a magnitude of 4,3 and that of vector B is 3,5, calculate the angle between these two vectors. We will find the angle using the above-mentioned formula.

\[\cos \theta = \frac{{A.B}}{{|A||B|}}\]

Firstly, we will find the magnitude of the vectors by using the following method:

|A| = √(4)2 + (3)2
|A| = √25
|A| = 5

For that of vector b, using the same method we will get the magnitude is √34

Secondly, we will find the dot product of the vectors

A.B = 4.3 + 3.5
A.B = 12 + 15
A.B = 27

Putting the values in the formula, we get:

\[\cos \theta = \frac{{A.B}}{{|A||B|}}\] \[\cos \theta = \frac{{27}}{{5\sqrt {34} }}\]

θ = 22.17°

Use of Angle between Two Vectors Calculator

This calculator can be helpful in getting the results in a few seconds. You simply have to put the values against the boxes and the result will be printed in the output field.