# Cross Product Calculator

The cross product of the two arrows is defined as the third arrow that is perpendicular to the two original arrows.

## What is Cross Product Calculator?

'Cuemath's Cross Product Calculator' is an online tool that helps to calculate the cross product of the given two vectors. Cuemath's online Cross Product Calculator helps you to calculate the cross product in a few seconds.

## How to Use Cross Product Calculator?

Please follow the below steps to calculate the cross product:

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

## How to Find Cross Product 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 = a _{1}i^ + a_{2}j^ + a_{3}k^**

**b = b _{1}i^ + b_{2}j^ + b_{3}k^**

Where a_{1}, b_{1}, c_{1,} and a2, b_{2}, c_{2 }are numeric values and i^, j^, k^ are the unit vectors along the x-axis, y-axis, and z-axis respectively.

When two vectors are multiplied with each other and the product is also a vector quantity, then the resultant vector is called the **cross product** or the vector product. Cross product is represented by a × b. The cross product of two vectors is given by:

**Cross product(a × b) = i(a _{2}b_{3 }− a_{3}b_{2}) − j(a_{1}b_{3 }− a_{3}b_{1}) + k(a_{1}b_{2 }− a_{2}b_{1})**

**Solved Example:**

Find the cross product of two vectors a = 4i + 2j – 5k and b = 3i – 2j + k?

**Solution:**

Given a = 4i + 2j – 5k and b = 3i – 2j + k

a × b = i(a_{2}b_{3 }− a_{3}b_{2}) − j(a_{1}b_{3 }− a_{3}b_{1}) + k(a_{1}b_{2 }− a_{2}b_{1})

= i((2 × 1) - (-5) × (-2)) - j((4 × 1 - (5) × (3)) + k((4) × (-2) - (2 × 3))

= i(2 - 10) - j(4 + 15) + k(-8 - 6)

= -8i - 19j - 14k

Therefore, the cross product of two vectors is -8i - 19j - 14k

Similarly, you can use the calculator to find the cross product of two vectors for the following:

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