Cross Product Calculator

vector a = i + j + k
vector b = i + j + k

Cross Product Lesson

Lesson Contents

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. A vector with one or two components of zero may still be considered to occupy three-dimensional space.

The cross product creates a resultant vector that 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 on which both original vectors lie on. Essentially, a cross product is a description of the parallelogram created by two vectors on their common plane.

what is a cross product
   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

cross product formula

        The cross product of a = u1i + u2j + u3k, and b = v1i + v2j + v3k

How the Calculator Works

The cross product calculator is written in the programming language JavaScript (JS). Your device’s internet browser has a built in JS engine that runs the code immediately when you click the calculate button. This allows instant solutions and no waiting on the page to refresh.

The core routine of the calculator uses the same process as shown in the image above. Your inputted vector components are filtered, built into an array (which is effectively a vector in JS), and then fed to the cross product equations. The resultant components are then built into the resultant vector.

The resultant vector is converted to LaTeX (a math rendering language) and then displayed in the answer area. Sometimes the inputted components do not have a solution or an error occurs during calculations. In the case of these anomalies, an error message will be displayed in the answer area.

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