Cross Product Calculator

This vector multiplication calculator aka cross multiply calculator helps to find the resultant vector of two given vectors. You can click on the “show more” option to see the step-by-step solution.

This vector calculator allows you to input information in the form of coordinates as well as points of the vector. 

What is the cross-product of two vectors?

Vectors can be multiplied to find the resultant vector. There are two ways to multiply a pair of vectors. 

• Scalar or dot product (resulting quantity is scalar).
• Vector or cross product (resulting quantity is a vector).

Cross product is defined as:

“Cross products only work in 3D. It measures how much two vectors point in different directions.”

It is represented by A x B (read as A cross B).

Where,

A x B = A*B sin 

Cross-product formula

 The formula used for vector cross product is a little complex. Firstly, the vectors are written in the form of a matrix. The first row of the matrix is of unit vectors.

 i      j      k

 a_x   a_y   a_z

 b_x   b_y   b_z

After this step, this matrix is expanded.

Properties of cross-product

There are certain properties of cross-product which differ it from the dot product.

• The commutative property is not fulfilled (i.e A x B ≠ B x A).
• It is maximum when the vectors are perpendicular (angle is 90).
• Self cross product results in a zero vector (i.e A x A = 0). 
• The Cross product of two unit vectors results in the third unit vector.(i x j = k, j x k = i, k x i = j)

How to do cross product?

The process of vector multiplication can be more easily understood through an example.

Example:

Find the cross product of the following vectors.

A = 3i + 2j + 1k

B = 1i + 2j + 3k

Solution:

Step 1: Write vectors in the form of coordinates.

A = (3,2,1)

B = (1,2,3)

Step 2: Form the matrix.

i       j       k

3      2      1

1      2      3

Step 3: Expand the matrix.

= i[(2).(3) - (1).(2)] - j[(3).(3) - (1).(1)] + k[(3).(2) - (2).(1)]           

= i[(6) - (2)] - j[(9) - (1)] + k[(6) - (2)]

= 4i - 8j + 4k