# Cross Product Calculator

Calculate the cross product of two vectors using the calculator below. See the steps to solve with the solution below.

## Cross Product of Vectors (a × b):

### Steps to Solve

#### Use the Cross Product Formula

(x_{a}, y_{a}, z_{a}) × (x_{b}, y_{b}, z_{b}) = ({y_{a}z_{b} - z_{a}y_{b}}, -{x_{a}z_{b} - z_{a}x_{b}}, {x_{a}y_{b} - y_{a}x_{b}})

#### Substitute Values and Solve

Enter vectors a & b above to see the solution here

## On this page:

## How to Calculate the Cross Product of Two Vectors

A cross product is the product between two vectors *a* and *b* in a three-dimensional space. The resulting vector is perpendicular to both of the vectors *a* and *b*.

The cross product is denoted *a × b*, and it differs from the dot product in that the result will be a vector rather than a number.

### Cross Product Formula

You can calculate the cross product of two vectors using the given formula:

a × b = |a|·|b|·sin(θ)·n

Where:

*|a|*= magnitude of vector*a**|b|*= magnitude of vector*b**θ*= angle between the vectors*n*= unit vector perpendicular to plane containing*a*and*b*

You can use our magnitude and angle between two vectors calculators to solve for *|a|*, *|b|*, and *θ*.

#### Practical Application

You can use an alternative formula to reduce the complexity of calculating the cross product in a three-dimensional space.

(x_{a}, y_{a}, z_{a}) × (x_{b}, y_{b}, z_{b}) = ({y_{a}z_{b} – z_{a}y_{b}}, -{x_{a}z_{b} – z_{a}x_{b}}, {x_{a}y_{2} – y_{a}x_{b}})

To use the formula, simply substitute the values of two vectors for x_{a}, y_{a}, z_{a}, x_{b}, y_{b}, & z_{b} to solve the resulting vector.

So, to solve x, y, and z for vector c, follow these steps:

x_{c} = {y_{a}z_{b} – z_{a}y_{b}}

y_{c} = -{x_{a}z_{b} – z_{a}x_{b}}

z_{c} = {x_{a}y_{2} – y_{a}x_{b}}

You might also be interested in our vector addition and vector subtraction calculators.