3D Cross Product Calculator

3D Cross Product Formula:

\[ \mathbf{A} \times \mathbf{B} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ A_x & A_y & A_z \\ B_x & B_y & B_z \end{vmatrix} = (A_yB_z - A_zB_y)\mathbf{i} - (A_xB_z - A_zB_x)\mathbf{j} + (A_xB_y - A_yB_x)\mathbf{k} \]

Cross Product (A × B):

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1. What is the 3D Cross Product?

The cross product is a binary operation on two vectors in three-dimensional space that results in a vector perpendicular to both original vectors. It's widely used in physics, engineering, and computer graphics.

2. How Does the Calculator Work?

The calculator uses the standard cross product formula:

\[ \mathbf{A} \times \mathbf{B} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ A_x & A_y & A_z \\ B_x & B_y & B_z \end{vmatrix} \]

Which expands to:

\( x \)-component: \( A_yB_z - A_zB_y \)
\( y \)-component: \( -(A_xB_z - A_zB_x) \)
\( z \)-component: \( A_xB_y - A_yB_x \)

Properties: The resulting vector is perpendicular to both A and B, and its magnitude equals the area of the parallelogram formed by A and B.

3. Applications of Cross Product

Details: The cross product is essential for calculating torque, angular momentum, surface normals in 3D graphics, and determining if two vectors are parallel.

4. Using the Calculator

Tips: Enter the x, y, z components of both vectors. The calculator will compute the resulting vector components.

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between dot product and cross product?
A: Dot product gives a scalar quantity measuring projection, while cross product gives a vector quantity measuring perpendicularity.

Q2: What does a zero cross product mean?
A: A zero cross product indicates the vectors are parallel (or at least one is zero).

Q3: Is the cross product commutative?
A: No, A × B = - (B × A). It's anti-commutative.

Q4: Why is the cross product only defined in 3D?
A: The cross product as commonly defined only works in 3D. In 7D there's a similar operation, and in other dimensions different operations are used.

Q5: How is cross product used in computer graphics?
A: It's used to calculate surface normals for lighting calculations and to determine polygon orientation.