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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.

Table of Contents:

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** = 3**i** + 2**j **+ 1**k**

**B** = 1**i** + 2**j **+ 3**k**

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**