# Angle Between Two Vectors Calculator

**Angle between two vectors** is the shortest **angle** at which any of the **two vectors** is rotated about the other **vector** such that both of the **vectors** have the same direction.

## What is Angle Between Two Vectors Calculator?

'Cuemath's Angle Between Two Vectors Calculator' is an online tool that helps to calculate the angle between two vectors. Cuemath's online Angle Between Two Vectors Calculator helps you to calculate the angle between two vectors in a few seconds.

## How to Use Angle Between Two Vectors Calculator?

Please follow the below steps to calculate the angle between two vectors:

**Step 1:**Enter the coefficients of two vectors in the given input boxes.**Step 2:**Click on the**"Calculate"**button to calculate the angle between two vectors.**Step 3:**Click on the**"Reset"**button to clear the fields and enter the different values.

## How to Find Angle Between Two Vectors Calculator?

Vectors are quantities with both magnitude and direction. Vectors help to simultaneously represent different quantities in the same expression.

The standard form of representation of a vector is:

**A = ai^ + bj^ + ck^**

Where, a, b, c are numeric values, and i^, j^, k^ are the unit vectors along the x-axis, y-axis, and z-axis respectively.

To calculate the angle between the vectors using the following formula. We know that the dot product of the vectors is given by

**a.b = |a||b|cosθ ⇒ cosθ = a.b / |a||b|**

Where a and b are vectors, θ is the angle between the vectors. |a|, |b| are the magnitude of both vectors.

**Solved Example:**

Find the angle between the two vectors 2i + j – 3k and 3i – j + k?

**Solution:**

Given a = 2i + j – 3k and b = 3i – j + k

cosθ = a.b / |a||b|

a.b = (2i + j – 3k).(3i – j + k)

= (2 × 3) + (1 × -1) + (-3 × 1)

= 6 - 1 - 3

= 2

|a| = √2^{2} + 1^{2} + (-3)^{2}

= √4 + 1 + 9

= √14

|b| = √3^{2} + (-1)^{2} + (1)^{2}

= √9 + 1 + 1

= √11

cosθ = 2 / (√14 × √11)

cosθ = 2 / 12.409

cosθ = 0.161

θ = cos^{-1}(0.161)

θ = 80.73°

Similarly, you can use the calculator to find the angle between two vectors for the following:

- a = 4i + 2j - 5k and b = -1i + 4j - 3k
- a = -2i - 5k and b = -7i + j + k