Calculation of Cross Product

Cross Product Formula:

\[ \mathbf{A} \times \mathbf{B} = (A_yB_z - A_zB_y)\mathbf{i} - (A_xB_z - A_zB_x)\mathbf{j} + (A_xB_y - A_yB_x)\mathbf{k} \]

Vector A x-component:

Vector A y-component:

Vector A z-component:

Vector B x-component:

Vector B y-component:

Vector B z-component:

Cross Product Result (A × B):

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1. What is the Cross Product?

The cross product is a binary operation on two vectors in three-dimensional space. It results in a vector that is perpendicular to both original vectors, with magnitude equal to the area of the parallelogram that the vectors span.

2. How Does the Calculator Work?

The calculator uses the standard cross product formula:

\[ \mathbf{A} \times \mathbf{B} = (A_yB_z - A_zB_y)\mathbf{i} - (A_xB_z - A_zB_x)\mathbf{j} + (A_xB_y - A_yB_x)\mathbf{k} \]

Where:

\( \mathbf{A} = (A_x, A_y, A_z) \) — First vector components
\( \mathbf{B} = (B_x, B_y, B_z) \) — Second vector components
\( \mathbf{i}, \mathbf{j}, \mathbf{k} \) — Unit vectors in x, y, z directions

Explanation: The cross product produces a vector perpendicular to both input vectors, with magnitude proportional to the sine of the angle between them.

3. Importance of Cross Product

Details: The cross product is essential in physics for calculating torque, angular momentum, and electromagnetic fields. In computer graphics, it's used for calculating surface normals.

4. Using the Calculator

Tips: Enter all six components (x, y, z for both vectors). The result will be a vector in the format (i, j, k) components.

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between dot product and cross product?
A: Dot product gives a scalar quantity, while cross product gives a vector quantity perpendicular to both input vectors.

Q2: What does a zero cross product mean?
A: A zero cross product indicates that the vectors are parallel (or one/both are zero vectors).

Q3: Can cross product be calculated in 2D?
A: In 2D, the cross product is a scalar (the z-component of what would be the 3D result).

Q4: What's the right-hand rule?
A: Point fingers in direction of first vector, curl towards second vector - thumb points in direction of cross product.

Q5: Is cross product commutative?
A: No, A×B = -B×A (anti-commutative).