# Cross Product Calculator

Cross Product Calculator calculates the cross product of the given two vectors. The cross product of the two vectors results in a third vector that is perpendicular to the two original vectors. Further, the direction of the vector obtained after taking the cross-product of two vectors can be determined by the right-hand rule.

## What is Cross Product Calculator?

Cross Product Calculator is an online tool that computes the cross product of two vectors. If two vectors are either in the same or opposite direction then their cross product is zero. Moreover, if any vector has zero length then the cross-product will again be zero. To use the * cross product calculator* enter the input values in the boxes.

### Cross Product Calculator

*Use 2 digits only.

## How to Use Cross Product Calculator?

Please follow the steps below to find the cross product using an online cross product calculator:

**Step 1:**Go to Cuemath’s online cross product calculator.**Step 2:**Enter the coefficients of two vectors in the given input boxes of the cross product calculator.**Step 3:**Click on the**"Calculate"****Step 4:**Click on the**"Reset"**

## How Does Cross Product Calculator Work?

**Vectors** are quantities with both magnitude and direction. Vectors help to simultaneously represent different quantities in the same expression. A number of arithmetic operations can be performed on vectors such as addition, subtraction, and multiplication. There are two types of multiplications performed on vectors. These are the dot product and the cross product.

The standard form of representation of a vector is:

a = \(a_{1}i\hat{}+a_{2}j\hat{}+a_{3}k\hat{}\)

b = \(b_{1}i\hat{}+b_{2}j\hat{}+b_{3}k\hat{}\)

Where \(a_{1}\), \(a_{2}\), \(a_{3}\) and \(b_{1}\), \(b_{2}\), \(b_{3}\) are numeric values. \(i\hat{}\), \(j\hat{}\) and \(k\hat{}\) are the unit vectors along the x-axis, y-axis, and z-axis respectively.

When two vectors are multiplied with each other and the product is also a vector quantity, then the resultant vector is called the **cross product** or the vector product. Cross product is represented by a × b. The cross product of two vectors is given by:

a × b = \({\begin{bmatrix} i\hat{} &j\hat{} &k\hat{} \\ a_{1} & a_{2} & a_{3} \\ b_{1} & b_{2} & b_{3} \end{bmatrix}}\)

If we use the cofactor expansion to solve this matrix the cross product is given as follows:

(a × b) = (\(a_{2}\)\(b_{3}\)_{ }− \(a_{3}\)\(b_{2}\))\(i\hat{}\) − (\(a_{1}\)\(b_{3}\)_{ }− \(a_{3}\)\(b_{1}\))\(j\hat{}\) + (\(a_{1}\)\(b_{2}\)_{ }− \(a_{2}\)\(b_{1}\))\(k\hat{}\).

## Solved Examples on Cross Product

**Example 1:** Find the cross product of two vectors a = 4\(i\hat{}\) + 2\(j\hat{}\) – 5\(k\hat{}\) and b = 3\(i\hat{}\) – 2\(j\hat{}\) + \(k\hat{}\) and verify it using cross product calculator.

**Solution:**

Given a = 4\(i\hat{}\) + 2\(j\hat{}\) – 5\(k\hat{}\) and b = 3\(i\hat{}\) – 2\(j\hat{}\) + \(k\hat{}\)

(a × b) = (\(a_{2}\)\(b_{3}\)_{ }− \(a_{3}\)\(b_{2}\))\(i\hat{}\) − (\(a_{1}\)\(b_{3}\)_{ }− \(a_{3}\)\(b_{1}\))\(j\hat{}\) + (\(a_{1}\)\(b_{2}\)_{ }− \(a_{2}\)\(b_{1}\))\(k\hat{}\).

= (2 - 10)\(i\hat{}\) - (4 + 15)\(j\hat{}\) + (-8 - 6)\(k\hat{}\)

= -8\(i\hat{}\) - 19\(j\hat{}\) - 14\(k\hat{}\)

Therefore, the cross product of two vectors is -8\(i\hat{}\) - 19\(j\hat{}\) - 14\(k\hat{}\)

**Example 2:** Find the cross product of two vectors a = 3\(i\hat{}\) + 6\(j\hat{}\) – 5\(k\hat{}\) and b = 5\(i\hat{}\) – 8\(j\hat{}\) + \(k\hat{}\) and verify it using cross product calculator.

**Solution:**

Given a = 3\(i\hat{}\) + 6\(j\hat{}\) – 5k and b = 5\(i\hat{}\) – 8\(j\hat{}\) + \(k\hat{}\)

(a × b) = (\(a_{2}\)\(b_{3}\)_{ }− \(a_{3}\)\(b_{2}\))\(i\hat{}\) − (\(a_{1}\)\(b_{3}\)_{ }− \(a_{3}\)\(b_{1}\))\(j\hat{}\) + (\(a_{1}\)\(b_{2}\)_{ }− \(a_{2}\)\(b_{1}\))\(k\hat{}\).

= (6 - 40)\(i\hat{}\) - (3 + 25)\(j\hat{}\) + (-24 - 30)\(k\hat{}\)

= -34\(i\hat{}\) - 28\(j\hat{}\) - 54\(k\hat{}\)

Therefore, the cross product of two vectors is -34\(i\hat{}\) - 28\(j\hat{}\) - 54\(k\hat{}\)

Similarly, you can use the cross product calculator to find the cross product of two vectors for the following:

- a = 4\(i\hat{}\) + 2\(j\hat{}\) - 5k and b = -1\(i\hat{}\) + 4\(j\hat{}\) - 3\(k\hat{}\)
- a = -2\(i\hat{}\) - 5k and b = -7\(i\hat{}\) + \(j\hat{}\) + \(k\hat{}\)

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