# Cross Product Calculator

A cross product calculator, sometimes referred to as the cross product of two vectors calculator, is used to find the cross product of two vectors. For example, you can quickly and easily find the cross product of two vectors with the Mathematics Master free online calculator.

## What is a Cross Product

The cross product, sometimes referred to as the vector product in mathematics, is a binary operation of two vectors in three dimensions. The “Cross Product” results from the two vectors, a and b. It can only be expressed in three dimensions and is symbolized by “a b.” The product of the two vectors, denoted by the letter “c,” is perpendicular to both vectors a and b.

## Cross Product Formula

The formula to calculate the cross products:

$$a × b = |a| |b| sin (θ) n$$

Where,

• $$| a |$$ and $$| b |$$ denote the vector lengths
• θ is the vectors’ angle (0° to 180°).
• n denotes the unit vector perpendicular to vectors a and b.
• $$| \space |$$ displays the magnitude

## Cross Product of Two Vectors

Cross product is a type of vector multiplication carried out between two vectors of various forms or natures. For example, the cross product and dot product can be used to multiply two or more vectors. The resultant vector is the cross-product of two vectors or the vector product.

## Properties of Cross product

The cross product has four properties, and the first two are the most crucial. These are:

### 1- Anti-commutative Property

This property primarily exhibits negative signs.

$$\overrightarrow{A} × \overrightarrow{B} = -\overrightarrow{B} × \overrightarrow{A}$$

The anti-commutativity takes part in the coordinate notation method, so the equalities i, j and k become -i, -j, and -k, respectively.

### 2- Distributive Property

It demonstrates how to resolve equations like a $$(b + c)$$.

$$\overrightarrow{A} × (\overrightarrow{B} + \overrightarrow{C}) = \overrightarrow{A} × \overrightarrow{B} + \overrightarrow{A} × \overrightarrow{C}$$

### 3- Jacobi Property

Jacobi property, named after the German Mathematician Carl Gustav Jakob, is represented.

$$\overrightarrow{A} × (\overrightarrow{B} × \overrightarrow{C}) + \overrightarrow{B} × (\overrightarrow{C} × \overrightarrow{A}) + \overrightarrow{C} × (\overrightarrow{A} × \overrightarrow{B}) = 0$$

### 4- Zero Vector Property

Represented as:

$$a × b = 0$$ if $$a = 0$$ or $$b = 0$$

There are three distinct sub-properties to it.

1. An identity that is additive, i.e., a + 0 Equals a.
2. Zero multiplied by any spatial vector has this feature; hence 0 (a) = 0.
3. Every vector in space is at right angles (orthogonal), which means that a. 0 = 0.

## FAQs

What is a vector cross-product calculator?

Vector cross product calculator is used for calculating the cross product between two vectors. It functions as a vector multiplication calculator.

Why is Cross Product Sine?

Since the cross product of two vectors determines the area of the parallelogram, sine is used because it is the angle between the two original vectors.