Mathematical Calculators

Vector Cross Product Calculator

Vector cross product calculator finds the cross product of two vectors in a three-dimensional space.

Vector A

Vector B

Vector C = A × B

Table of contents

Cross Product Calculation Formula
Definition of Cross Product
How to calculate cross product of two vectors
What cross product is?
To determine the cross product of a new vector, you need to enter the x, y, and z values of two vectors into the calculator.

Cross Product Calculation Formula

The formula for calculating the new vector of the cross product of two vectors is the following:
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.

Definition of Cross Product

A cross product, which is also known as a vector product, is a mathematical operation. In cross product operation 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 should not to be confused with the dot product. The dot product is a simpler algebraic operation that returns a single number as opposed to a new vector.

How to calculate cross product of two vectors

Here is an example of calculating the cross-product for two vectors.
First thing is to gather two vectors: vector A and vector B. For this example, we will assume vector A has coordinates of (2, 3, 4) and vector B has coordinates of (3, 7, 8).
After this we use the simplified equation above to calculate the resulting vector coordinates of the cross product.
Our new vector will be denoted as 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 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 cross product is?

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

John Cruz
Article author
John Cruz
John is a PhD student with a passion to mathematics and education. In his freetime John likes to go hiking and bicycling.

Vector Cross Product Calculator English
Published: Sun Jul 04 2021
In category Mathematical calculators
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