Cross Product Calculator – Cross Product of Two Vectors (Vector Product)

The Cross Product Calculator calculates the cross product of two given vectors, a and b. To find the cross product, enter the x,y, and z values of two vectors into the calculator below to compute the resulting cross product.

How to use the cross product calculator

This vector multiplication calculator is simple and easy to use. Instead of manual computations, it provides you with the vector cross product in a matter of seconds. Use our online vector product calculator to find the cross products.

1. Input the 3 values for vector a (x, y, z).
2. Input the 3 values for vector b (x, y, z).
3. After you've entered all the values, you'll get the solution, the cross products of vector c (x, y, z)

Cross product formula

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

a × b =‖a‖ ‖b‖ sin(θ) n

where:

θ is the angle between a and b in the plane containing them (between 0 – 180 degrees)

‖a‖ and ‖b‖ are the magnitudes of vectors a and b

n is the unit vector perpendicular to a and b

Vector coordinates

In terms of vector coordinates, the above formula can be simplified as follows:

a × b = (a₂b₃ - a₃b₂, a₃b₁ - a₁b₃, a₁b₂ - a₂b₁)

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.

The vectors in three dimensions are: i = (1, 0, 0), j = (0, 1, 0), k = (0, 0, 1)

From the definition, these rules apply:

i × j = k j × i = i k × i = j

j × i = -k k × j = -i i × k = -j

i × i = j × j = k × k = 0

the cross product of both vectors will be:

a × b = (a₁i + a₂j + a₃k) × (b₁i + b₂j + b₃k)

Example of how to calculate the cross product of two vectors

First, we need two vectors a and b
Given:
vector a has coordinates of (2,1,4) and
vector b has coordinates of (3,1,2).

Second, use the simplified equation above to calculate the resulting vector coordinates of the cross product.

a × b = (a₂b₃ - a₃b₂, a₃b₁ - a₁b₃, a₁b₂ - a₂b₁)

We need to find the x coordinate. Through the formula above, we find x to be 1*2 - 4*1 = 2 - 4 = -2.

Using the same equation, we find y and z to be 8 and -1 respectively.

Finally, we find the new vector from the cross product of a × b is (-2, 8, -1)

Important: Don't forget that in a cross product, the result of a × b is not the same as b × a. In fact a × b = -b × a.

FAQs

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 definition

A cross product (aka a vector product) is the result of a cross product between two vectors, i.e., a new vector that is perpendicular to both vectors. The magnitude of the new vector is equal to the area of a parallelogram with the sides of the two original vectors.

How to calculate the cross product

The multiplication of vectors refers to the cross product, and it can be done in two of the following ways, depending on which input you know.

1. Magnitude of Vector A and B

A cross product tells you how different dimensions are interacting with each other. If you know the magnitude of the vector a and b, then you can compute the magnitude by multiplying them with a sine angle forming between both vectors.

∥a∥=√(a₁²+a₂²+a₃²)

a × b =‖a‖ ‖b‖ sin(θ) n

2. Right-hand Rule

We can also find the direction of a cross-product vector using the right-hand rule. Place your index finger, index finger and thumb so that they all form up like an X, Y, and Z coordinate system. Then move your hand to the point that your index finger points to vector a and your middle finger points to vector b. Your thumb will indicate the direction of the cross product vector.