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Cross Product Calculator

To find the cross product of two vectors, enter the coordinates or points of both vectors in the cross product calculator.

First vector


Second vector


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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

 ax   ay   az

 bx   by   bz

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