Enter the x,y, and z values of two vectors into the calculator below to determine the cross product as a new vector.

## What is a cross product?

A cross product, also known as a vector product, is a mathematical operation in which the result of the cross product between 2 vectors is a new vector that is perpendicular to both vectors. The magnitude of this new vector is equal to the area of a parallelogram with sides of the 2 original vectors.

The cross product is not to be confused with the dot product which is a simpler algebraic operation that returns a single number as opposed to a new vector.

## Cross Product Formula

The formula for calculating the new new vector of the cross product of two vectors is as follows:

- Where
*θ*is the angle between a and b in the plane containing them. (Always between 0 – 180 degrees) - ‖
**a**‖ and ‖**b**‖ are the magnitudes of vectors a and b - and n is the unit vector perpendicular to a and b

In terms of vector coordinates we can simply the above equation into the following:

a x b = (a2*b3-a3*b2, a3*b1-a1*b3, a1*b2-a2*b1)

Where a and b are vectors with coordinates (a1,a2,a3) and (b1,b2,b3).

The direction of the resulting vector can be determined with the right hand rule. This is done by

## How to calculate a cross product

The following is an example calculating the cross product of two vectors.

- First, lets gather our two vectors a and b. For this example we will assume vector a has coordinates of (2,3,4) and vector b has coordinates of (3,7,8).
- Next we must use the simplified equation above to calculate the resulting vector coordinates of the cross product.
- Our new vector will be denoted c, so first we will want to find the x coordinate. Through the formula above we find x to be -4.
- Using the same method we then find y and z to be .-4 and 5 respectively.
- Finally we having our new vector from the cross product of a X b of (-4,-4,5)

It’s important to remember that the cross product is anti commutative meaning that the the result of a X b is not the same as b X a. In fact a X b = -b X a.

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