Cross Product Calculator: Fast & Easy Vector Solution

Q: 5. What are some real-world applications of the cross product?

The cross product finds applications in various fields: Physics: Calculating torque, angular momentum, and magnetic force.Engineering: Determining moments of forces and analyzing structures.Computer graphics: Computing surface normals and handling 3D rotations.

Q: 8. Is the cross product commutative?

No, the cross product is not commutative. A × B ≠ B × A. In fact, A × B = - (B × A).

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How to Calculate the Cross Product of Two Vectors (Step-by-Step Guide)

Cross Product Calculator – Free Online Tool with Formula, Steps & Examples

Cross Product Calculator

What is Cross Product Calculator?

A Cross Product Calculator is an online mathematical tool that lets you quickly find the cross product (vector product) of two 3-dimensional (3D) vectors. The cross product gives a new vector that is perpendicular to both input vectors, used commonly in Physics, Engineering, and Geometry. Simply enter the components of vectors A and B, click calculate, and see the result vector, magnitude, and full working instantly.

Formula or Logic Behind Cross Product Calculator

The cross product of two vectors A and B in 3D space is calculated using the determinant formula:

A × B = (A_yB_z – A_zB_y, A_zB_x – A_xB_z, A_xB_y – A_yB_x)

Or using the matrix determinant method:

A × B = |i j k|
|A_x A_y A_z|
|B_x B_y B_z|

Where i, j, k are the unit vectors along the X, Y, Z axes. This combination gives a vector at right angles to both input vectors (Right-Hand Rule).

Solved Examples: Cross Product of Two Vectors

Vector A (A_x, A_y, A_z)	Vector B (B_x, B_y, B_z)	Cross Product (A × B)
(1, 0, 0)	(0, 1, 0)	(0, 0, 1)
(2, 3, 4)	(5, 6, 7)	(-3, 6, -3)
(0, 1, 2)	(3, 5, 6)	(-4, 6, -3)
(3, 1, 4)	(2, 0, 5)	(5, -7, -2)

Steps to Use the Cross Product Calculator

Enter the required number or values: Input the 3 components of each vector (A and B).
Click on the 'Calculate' button.
Get instant results: The cross product vector and step-by-step explanation will be shown.

Why Use Vedantu’s Cross Product Calculator?

Vedantu’s Cross Product Calculator is designed for accuracy and ease-of-use. It gives instant solutions, shows detailed steps, and works seamlessly on mobile and desktop. Trusted by students, teachers, and professionals, it’s based on standard NCERT and CBSE/ICSE math concepts and perfect for exam prep or practical problem-solving.

Real-life Applications of Cross Product Calculator

The cross product, and this calculator, have widespread real-life uses: computing torque in physics, finding normal vectors for 3D geometry in engineering and computer graphics, robotics navigation, finding the area of parallelograms, and determining perpendicular directions in maps and navigation. It’s also helpful for fast solutions in classroom assignments, JEE/NEET competitive exams, and science lab work.

For more on related topics, see our guides on Dot Product (Scalar Product) Calculator or brush up with Vector Algebra Notes. Learn about matrices on Matrices in Maths and practice with Algebra Topics.

FAQs on Cross Product Calculator: Fast & Easy Vector Solution

1. What is a cross product calculator?

A cross product calculator is an online tool that computes the cross product of two three-dimensional vectors. It simplifies the process by performing the calculations automatically, providing both the resulting vector and a step-by-step explanation.

2. How do I calculate the cross product of two vectors?

The cross product of two vectors, A and B, is calculated using the determinant of a 3x3 matrix. The formula is: A × B = |i j k| |Ax Ay Az| |Bx By Bz|. This expands to a vector with components: (AyBz - AzBy)i - (AxBz - AzBx)j + (AxBy - AyBx)k. A cross product calculator automates this complex calculation.

3. What is the cross product formula?

4. What does the cross product of two vectors represent?

The cross product of two vectors results in a new vector that is perpendicular to both of the original vectors. Its magnitude represents the area of the parallelogram formed by the two vectors. The direction is determined by the right-hand rule.

5. What are some real-world applications of the cross product?

The cross product finds applications in various fields:

Physics: Calculating torque, angular momentum, and magnetic force.
Engineering: Determining moments of forces and analyzing structures.
Computer graphics: Computing surface normals and handling 3D rotations.

6. How do I use a cross product calculator?

Simply input the components (x, y, z) of each vector into the designated fields on the calculator. Click ‘Calculate’ to obtain the resultant cross product vector and a step-by-step solution.

7. What is the difference between the dot product and the cross product?

The dot product results in a scalar (single number) representing the projection of one vector onto another. The cross product, however, yields a vector perpendicular to both input vectors, representing the area of the parallelogram they form.

8. Is the cross product commutative?

No, the cross product is not commutative. A × B ≠ B × A. In fact, A × B = - (B × A).

9. What is the magnitude of the cross product?

The magnitude of the cross product of vectors A and B is given by ||A × B|| = ||A|| ||B|| sin θ, where θ is the angle between the two vectors. This represents the area of the parallelogram formed by A and B.

10. How is the right-hand rule used with the cross product?

The right-hand rule determines the direction of the resulting cross product vector. If you curl the fingers of your right hand from vector A to vector B, your thumb will point in the direction of the cross product A × B.

11. What if I enter vectors that are parallel?

If the input vectors are parallel (or anti-parallel), their cross product will be the zero vector (0, 0, 0). This is because the sine of the angle between parallel vectors is zero.

12. Can I use this calculator for any type of vector?

This calculator is specifically designed for three-dimensional vectors. It cannot be used for two-dimensional vectors or vectors in higher dimensions.