Unleash Vector Power: 3D Cross Product Calculator

Visualize and calculate the cross product of two 3D vectors in real-time. Interactive, intuitive, and perfect for all your vector math needs.

Enter Your Vectors

Define vectors A and B by entering their x, y, and z components.

Vector A

Component x (i)

Component y (j)

Component z (k)

Vector B

Component x (i)

Component y (j)

Component z (k)

Cross Product Result

Formula:

$$ \mathbf{a} \times \mathbf{b} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ a_x & a_y & a_z \\ b_x & b_y & b_z \end{vmatrix} = (a_y b_z - a_z b_y)\mathbf{i} - (a_x b_z - a_z b_x)\mathbf{j} + (a_x b_y - a_y b_x)\mathbf{k} $$

Result Vector:

Understanding Cross Product

The cross product, also known as the vector product, is a fundamental operation in vector algebra, specifically in three-dimensional space. Given two vectors, it produces a third vector that is perpendicular to both of them.

Magnitude: The length of the cross product vector is equal to the area of the parallelogram formed by the two original vectors. It's calculated as $ |\mathbf{a} \times \mathbf{b}| = |\mathbf{a}| |\mathbf{b}| \sin(\theta) $, where $ \theta $ is the angle between vectors $\mathbf{a}$ and $\mathbf{b}$.

Direction: The direction of the cross product is perpendicular to both input vectors, determined by the right-hand rule. If you curl the fingers of your right hand from vector $\mathbf{a}$ to vector $\mathbf{b}$, your thumb points in the direction of $\mathbf{a} \times \mathbf{b}$.

Applications: Cross product is crucial in physics and engineering for:

Torque Calculation: In mechanics, torque is calculated using the cross product of force and lever arm.
Magnetic Force: In electromagnetism, the force on a moving charge in a magnetic field is given by a cross product.
Surface Normals: In computer graphics and 3D geometry, cross products are used to find surface normals for lighting and shading.

For further reading, visit Wikipedia on Cross Product.