# Cross Product Calculator

## Find the cross product of vectors step by step

The online calculator will find the cross product of two vectors, with steps shown.

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If the calculator did not compute something or you have identified an error, or you have a suggestion/feedback, please write it in the comments below.

The Cross Product Calculator is an online tool that allows you to calculate the cross product (also known as the vector product) of two vectors. The cross product is a vector operation that returns a new vector that is orthogonal (perpendicular) to the two input vectors in three-dimensional space.

Our vector cross product calculator is the perfect tool for students, engineers, and mathematicians who frequently deal with vector operations in their work or study.

## How to Use the Cross Product Calculator?

• ### Input the First Vector

The first step involves entering the coordinates of the first vector into the designated input fields. The vector can be in 2D or 3D. For instance, you might enter a 3D vector as $\mathbf{\vec{u}}=\langle u_1,u_2,u_3\rangle$.

• ### Input the Second Vector

After entering the first vector, you proceed to input the second vector's coordinates in the provided fields. The second vector should be of the same dimension as the first. For a 3D vector, you could enter it as $\mathbf{\vec{v}}=\langle v_1,v_2,v_3\rangle$.

• ### Calculate

After inputting both vectors, you can then click the "Calculate" button. The cross product calculator will immediately compute and display the cross product of the two input vectors.

## Cross Product Formula

The vector cross product, often referred to as the cross product, uses the cross product formula. If the two vectors are $\mathbf{\vec{u}}=\langle u_1,u_2, u_3\rangle$ and $\mathbf{\vec{v}}=\langle v_1,v_2, v_3\rangle$, their cross product $\mathbf{\vec{u}}\times\mathbf{\vec{v}}$ can be represented as follows:

$$\mathbf{\vec{u}}\times\mathbf{\vec{v}}=\langle u_2v_3-u_3v_2,u_3v_1-u_1v_3,u_1v_2-u_2v_1\rangle$$

This formula yields a new vector that is perpendicular to the plane formed by vectors $\mathbf{\vec{u}}$ and $\mathbf{\vec{v}}$, following the right-hand rule.

## What is right-hand rule?

The right-hand rule is a convention used in mathematics, physics, and engineering to determine the direction of certain vectors. It's especially useful when working with the cross product of two vectors.

Here's how you can use the right-hand rule for the cross product:

• Stretch out your right hand flat with the palm facing up.

• Point your index finger in the direction of the first vector.

• Bend your middle finger towards your palm so it points in the direction of the second vector.

• Your thumb, when extended, points in the direction of the resulting cross product vector.

This gives you a visual way to remember that the vector produced by the cross product of two vectors is perpendicular to the plane formed by those two vectors. The thumb of your right hand points in the direction of the resulting cross product vector.

Remember, the right-hand rule follows a specific orientation: the first vector is represented by the index finger, the second vector by the middle finger, and the resulting cross product by the thumb. The order in which the vectors are crossed is important since reversing the order will reverse the direction of the resulting cross product vector.

The right-hand rule is often used in fields such as electromagnetism, rotational dynamics, and computer graphics to determine the direction of various quantities.

## Grasping the Concept of Vector Cross Product

The cross product is a binary operation that combines two vectors in three-dimensional space to produce a third vector which is orthogonal to the initial vectors. This vector product is significant in physics and engineering because it helps model phenomena such as torque, angular momentum, and electromagnetism.

The cross product of two vectors is always perpendicular to the plane in which the two vectors lie. Moreover, the magnitude (length) of this product vector is equal to the area of the parallelogram with the two vectors as sides.

The cross product calculator thus comes in handy in various practical scenarios, whether you're determining the area of a parallelogram in a vector space or calculating torque in physics. No need to do manual calculations; let our online calculator handle your vector cross product needs!

For example, let's calculate the cross product of two vectors using our online calculator.

Suppose we have the first vector $\mathbf{\vec{u}}=\langle 2,3,4\rangle$ and the second vector $\mathbf{\vec{v}}=\langle 5,6,7\rangle$. Insert these values into their respective fields and click "Calculate." The resulting cross product will be $\mathbf{\vec{u}}\times\mathbf{\vec{v}}=\langle -3,6,-3\rangle$.

Our cross product calculator provides an intuitive and seamless way to calculate the cross product of two vectors. Give it a try now!

## Why Choose Our Cross Product Calculator?

• ### Accurate

It uses the correct product formula, ensuring you get accurate results every time.

• ### Quick

The cross product calculation is instant, saving you time.

• ### Versatile

It can handle both 2D and 3D vectors.

• ### Free

This tool is 100% free to use.

• ### User-friendly interface

It is simple to use for everyone.

### FAQ

#### Can I calculate the cross product of 2D vectors with this calculator?

Most cross product calculators, including ours, primarily deal with 3D vectors as these are most common in practical scenarios. If you input 2D vectors, the third coordinate will be automatically set to zero. The result will always be in the form $\langle 0,0,n\rangle$. That's why for 2D vectors, the cross product is typically represented as a scalar rather than a vector.

#### Is the order of vectors important in the cross product?

Yes, the order matters. The cross product is not commutative, meaning $\mathbf{\vec{u}}\times\mathbf{\vec{v}}$ is not the same as $\mathbf{\vec{v}}\times\mathbf{\vec{u}}$. In fact, they are negatives of each other. This order is reflected in the right-hand rule.

#### Is the Cross Product Calculator free to use?

Yes, our Cross Product Calculator is completely free to use.