Cross Product Calculator

What is the Cross Product?

In mathematics, particularly in linear algebra and vector calculus, the cross product is an operation applied to two vectors in three-dimensional space. The result of the cross product is another vector that is perpendicular to both of the original vectors. This operation has significant applications in physics, engineering, and computer graphics, where direction and orientation are critical.

How to Calculate Cross Product?

The cross product of two vectors, \( \vec{A} \) and \( \vec{B} \), is denoted by \( \vec{A} \times \vec{B} \). The resulting vector, \( \vec{C} \), is orthogonal to both \( \vec{A} \) and \( \vec{B} \). Mathematically, the cross product is defined as:

\[ \vec{C} = \vec{A} \times \vec{B} = \begin{bmatrix} A_y \cdot B_z - A_z \cdot B_y \\ A_z \cdot B_x - A_x \cdot B_z \\ A_x \cdot B_y - A_y \cdot B_x \end{bmatrix} \]

Here, \( A_x, A_y, A_z \) and \( B_x, B_y, B_z \) represent the components of vectors \( \vec{A} \) and \( \vec{B} \) along the \( x \), \( y \), and \( z \) axes, respectively.

Geometric Interpretation of the Cross Product

The cross product not only provides a vector orthogonal to the original two vectors but also encodes additional information:

- The magnitude of \( \vec{A} \times \vec{B} \) is given by: \[ |\vec{A} \times \vec{B}| = |\vec{A}| |\vec{B}| \sin(\theta) \] where \( \theta \) is the angle between \( \vec{A} \) and \( \vec{B} \).

- The direction of \( \vec{A} \times \vec{B} \) is determined by the right-hand rule. If you point the fingers of your right hand in the direction of \( \vec{A} \), curl them towards \( \vec{B} \), your thumb points in the direction of the cross product.

A Practical Example

Let’s compute the cross product of two vectors:

\( \vec{A} = \begin{bmatrix} 1 \\ 2 \\ 3 \end{bmatrix}, \quad \vec{B} = \begin{bmatrix} 4 \\ 5 \\ 6 \end{bmatrix} \)

Using the formula: \[ \vec{A} \times \vec{B} = \begin{bmatrix} 2 \cdot 6 - 3 \cdot 5 \\ 3 \cdot 4 - 1 \cdot 6 \\ 1 \cdot 5 - 2 \cdot 4 \end{bmatrix} = \begin{bmatrix} -3 \\ 6 \\ -3 \end{bmatrix} \]

Thus, the cross product is: \( \vec{C} = \begin{bmatrix} -3 \\ 6 \\ -3 \end{bmatrix} \).

Cross Product Calculator: Simplify Your Calculations

Calculating the cross product by hand can be time-consuming and prone to errors, especially for large datasets or repetitive calculations. Our Cross Product Calculator provides a quick and accurate way to compute cross products for any two 3D vectors. Simply input the components of your vectors, and the calculator will return:

• The resulting vector (\( \vec{C} \))
• Intermediate steps (optional for learning purposes)

This tool is perfect for students, engineers, and professionals working in fields where vector calculations are essential.