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

Cross Product Formula

The formula for calculating the 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 simplify 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.

Cross Product Definition

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.

How to calculate a cross product

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

  1. First, let’s 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).
  2. Next, we must use the simplified equation above to calculate the resulting vector coordinates of the cross product.
  3. 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.
  4. Using the same method we then find y and z to be .-4 and 5 respectively.
  5. Finally, we having our new vector from the cross product of an X b of (-4,-4,5)

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


What is a cross product?

A cross product is a vector product that is perpendicular to both of the original vectors and is over the same magnitude.

cross product calculator
cross product formula