Cross Product Calculator

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

Vector a
Vector b

Cross Product of Vectors (a × b):

 

Steps to Solve

Use the Cross Product Formula

(xa, ya, za) × (xb, yb, zb) = ({yazb - zayb}, -{xazb - zaxb}, {xayb - yaxb})

Substitute Values and Solve

Enter vectors a & b above to see the solution here

Learn how we calculated this below


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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.

(xa, ya, za) × (xb, yb, zb) = ({yazb – zayb}, -{xazb – zaxb}, {xay2 – yaxb})

To use the formula, simply substitute the values of two vectors for xa, ya, za, xb, yb, & zb to solve the resulting vector.

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

xc = {yazb – zayb}
yc = -{xazb – zaxb}
zc = {xay2 – yaxb}

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