# Cross Product Calculator

In a vector product calculator, it's hard to understand the way to calculate the vector product. Luckily for you, we've made a tool that helps you understand the formula for the vector product of two vectors.

#### Cross Product Calculator

$$Vector \ A$$
$$Vector\ B$$

#### $$Cross\ Product\ Farmula$$

$$C=$$ $$a*b=|a|*|b|*Sinθ *n$$

#### Example:

$$\left(2, 13, -4\right)\times\left(-2, 15, 6\right)$$ To find the cross product, we form a determinant the first row of which is a unit vector, the second row is our first vector, and the third row is our second vector: $$\left | \begin{array}{ccc} \mathbf{i} & \mathbf{j} & \mathbf{k} \\\\ 2 & 13 & -4 \\\\ -2 & 15 & 6 \end{array} \right |$$ Now, just expand along the first row: $$\left | \begin{array}{ccc} \mathbf{i} & \mathbf{j} & \mathbf{k} \\\\ 2 & 13 & -4 \\\\ -2 & 15 & 6 \end{array} \right | =\left| \begin{array}{cc} 13 & -4 \\\\ 15 & 6 \end{array} \right|\mathbf{i}-\left| \begin{array}{cc} 2 & -4 \\\\ -2 & 6 \end{array} \right|\mathbf{j}+\left| \begin{array}{cc} 2 & 13 \\\\ -2 & 15 \end{array} \right|\mathbf{k}=$$ $$=(13 \cdot(6)-(15)\cdot(-4))\mathbf{i}-(2 \cdot(6)-(-2)\cdot(-4))\mathbf{j}+(2 \cdot(15)-(-2)\cdot(13))\mathbf{k}=$$ $$=138\mathbf{i}-4\mathbf{j}+56\mathbf{k}$$

$$\left(2, 13, -4\right)\times\left(-2, 15, 6\right)=\left(138, -4, 56\right)$$

## What is cross Product Calculator?

In a vector product calculator, it's hard to understand the way to calculate the vector product. Luckily for you, we've made a tool that helps you understand the formula for the vector product of two vectors.

We’ll even be comparing the scalar product vs cross-product definitions, and explain why they're not an equivalent operation. And as a bonus, we even have an inventory of practical tricks just like the right-hand rule, in order that you'll become a master on the way to do the vector product of two vectors.

## Vector product definition

A vector may be a mathematical tool widely utilized in physics. It allows you to affect collections of numbers (each representing a dimension) in a very efficient way. the gathering of operations, rules, and properties to affect vectors is named algebra and, similarly to the algebra of numbers, it includes multiplication.

However, vectors are more complex than numbers since they carry within them far more information that has got to be more carefully manipulated. this is often one of the explanations why, in algebra, there are two different types of multiplications or product operations: the vector product and therefore the scalar product.

The definition, because it is common in mathematics, is extremely technical. Nevertheless, we'll explain what it means in layperson's (and less accurate) terms in order that, albeit you do not have a robust mathematical background, everything will add up to you.

One definition of the vector product also called cross product is:

A binary operation on two vectors in three-dimensional space that's denoted by the symbol x. Given two linearly independent vectors, a and b, the vector product a × b, may be a vector that's perpendicular to both a and b and thus normal to the plane containing them.

That is indeed a mouthful, but we will translate it from mathematical jargon to an everyday explanation. First of all, the definition talks a few three-dimensional spaces.

Just like the one we sleep in because it's the foremost common usage of the vector product, but the vector product is often extended to more dimensions; that's, however, beyond the scope of this text and most math-related degrees.

Cross product formula

Before we present the formula for the cross product, we'd like two vectors that we'll call a and b. These two vectors shouldn't be collinear (a.k.a. shouldn't be parallel) for reasons that we'll explain afterwards.

So, without further ado, let's examine the formula:

c = a × b = |a| * |b| * sinθ * n

This formula consists of:

c - New vector resulting from doing the vector product,

a - one among the initial vectors,

b - Second of the initial vectors,

θ - Angle between both vectors,

n - Unit vector perpendicular to a and b simultaneously.

The factors of perpendicularity alongside the sinus function present within the formula are good indicators of the geometrical interpretations of the vector product. We’ll talk more about these within the next sections.

You can also see why it's crucial that the 2 vectors a and b aren't parallel. If they were parallel, it might cause a zero angle between them (θ = 0). Hence, both sin θ and c would be adequate to zero, which may be a very uninteresting result. Also interesting to notice is that the incontrovertible fact that an easy permutation of a and b would change only the direction of c since -sin (θ) = sin (-θ).

How to use this cross-product calculator

To use this cross-product calculator is really simple and also it’s a free and web-based tool that anyone can use.

This tool is really useful for a student to solve their problems. They don’t have to confuse about calculation. They don’t have to register or they don’t have to provide emails.

To use this tool you have to use this tool

You have to identify the value of the component of vectors a. That is: x = 2, y = 3 and `z = 7'.

Next, you should introduce the components of vector b. That is: x = 1, y = 2 and z = 4.

Now the calculator processes the information applies the formula we saw before and you have just calculated c = a × b = (-2, -1, 1)

Repeat again until you have calculated all the cross products you needed to.

Share with your friends the life-changing experience of calculating vector cross products. ;)

Tips: Also bookmark this tool so that you don’t have to search for this tool again and again.

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