Vector Cross Product Calculator

Find the cross-product of two vectors using this 3d (3x3) cross-product calculator. 

It also goes by the name matrix cross-product calculator since it uses the matrix cross-product method. 

If you already have vectors in the matrix form, input the first entry (a11) of the first row in the first i box, a12 in the j, and a13 in the k box of the cross-product calculator. Enter the values of the second row in the boxes designated for vector b.

How to use vector cross product calculator?

Enter the magnitudes of unit vectors i, j, and k of both vectors.
Click calculate.
To see the steps of the computation, click on “show more”.

What is a Vector cross product?

The vector cross product is a different type of product used in the fields of physics and math. It is performed between two, 3-dimensional vectors.

Cross product is represented as a x b. The result of this product is always a vector. It finds the difference between the directions of the vectors.

For example, The formula of torque is r x f. Here r is radius and f is force. Similarly, the force of a particle in a magnetic field is calculated using the vector product.

The formula of the cross product

There is also a mathematical formula to find the cross-product. That is:

A x B = |A||B| Sin θ

To use this formula, one needs the magnitudes of both vectors and the angle between them.

Right-hand thumb rule:

The resultant vector is always perpendicular to both vectors. The magnitude of this vector is equal to the parallelogram drawn using vectors a and b.

The direction of the resultant vector can be found by using the right-hand rule. It is 
   “Hold up your right hand. Open the index finger in the direction of vector a and your middle finger in the direction of vector b. The thumb points in the direction of the resultant vector.”

Properties of the Cross product:

1. Non-commutative product. A cross B is not equal to B cross A.

2. Two perpendicular vectors have the maximum magnitude of the cross product. It is because the value of sin θ at 90 degrees is 1. 

Unit vectors are an example of this case as they are mutually perpendicular. 
                       i x j = k,  j x k = i, k x i = j

3. Two parallel vectors have the minimum magnitude of cross-product. Because the angle between parallel vectors is 0 degrees and sin at 0^o and 180^o is 0.

4. Self cross-product is a null vector because the angle is 0 degrees.

Hence:
                        A x A = i x i = j x j = k x k = 0 

5. Cross-product in terms of the rectangular components.  
A x B = (Ax i + Ay j + Az k ) x (Bx i + By j + Bz k )
A x B = (AyBz - AzBy)i + (AzBx - AxBz)j + (AxBy - AyBx)k

 OR

How to find the vector cross product?

Depending on the type of the given data, you can choose between the two methods. If the magnitudes and angles are given, the formula method is the right choice.

But if the actual vectors are given, use the matrix cross-product method. Write the components of the vectors in a 3x3 matrix as shown in the properties section.

Then separate out the formal determinants by applying Sarrus’s rule of cofactor expansion and solve.

For more clarity, see the solved example below. 
Example:
Find the Vector cross product of the following vectors.
u = ( 3, -4, 5) and v = (8, 15, -17)

Solution:
Step 1: Write the vectors in matrix form.

Step 2: Expand using Sarrus’s rule.

= [(-4)(-17) - (5)(15)] i - [(3)(-17) - (5)(8)] j + [(3)(15) - (-4)(8)]k
= [68 -75] i - [-51 -40] j + [45 + 32] k
= -7i + 91j + 77k

The resultant vector is 
(-7, 92, 77)

If need be, verify using the vector cross-product calculator.