Programmers and Teachers:
Cross Product Calculator
Cross Product Lesson
What is a Cross Product?
The cross product of two vectors is a vector that is orthogonal to both original vectors. To calculate a cross product, the vectors that are being multiplied must occupy three-dimensional space.
The resultant vector is orthogonal to both original vectors. This means that a 90° angle can be drawn between the resultant vector and each of the original vectors. The magnitude of the resultant vector is equal to the area of the parallelogram that is projected from the two original vectors.
The image below is a visualization of what the direction and magnitude of a cross product represent. Note that the cross product is normal to the plane which both original vectors lie on.
A cross product tells us the normal direction
and projected area of two vectors.
How to Calculate the Cross Product
An easy way to remember how to calculate the cross product of two vectors is shown in the image below. If vector a = u1i + u2j + u3k, and vector b = v1i + v2j + v3k, then the multiplication matrix should be set up as shown, and the resultant vector will equal the summation shown on the left half of the image.
The summation is written in full form as:
a×b = (u2v3 – u3v2)i – (u1v3 – u3v1)j + (u1v2 – u2v1)k
The cross product of a = u1i + u2j + u3k, and b = v1i + v2j + v3k
How the Cross Product Calculator Works
The calculator on this page uses the same process as shown in the image above. Your inputs are sent to the calculator program and then it multiplies each component by its corresponding component.
After the multiplication is done, the i, j, and k components are added and subtracted. Once this simplification occurs, the components’ values are attached to the component letters. This final vector is sent back to this page and displayed to you.