3x3 Matrix Cross Product Calculator

Cross Product Formula:

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

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 (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 that results in a vector perpendicular to both original vectors. Its magnitude relates to the area of the parallelogram formed by the two vectors.

2. How Does the Calculator Work?

The calculator uses the determinant formula for cross product:

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

Where:

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

Calculation Steps:

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

3. Importance of Cross Product

Applications: The cross product is essential in physics (torque, angular momentum), engineering (moment of force), computer graphics (surface normals), and many other fields involving 3D vectors.

4. Using the Calculator

Instructions: Enter all six components (x, y, z for both vectors A and B). The calculator will compute the resulting vector perpendicular to both input vectors.

5. Frequently Asked Questions (FAQ)

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

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

Q3: Why is the cross product only defined in 3D?
A: The perpendicular vector concept only works in 3D. In 2D, the cross product is a scalar, and in higher dimensions, more complex operations are needed.

Q4: How is cross product related to area?
A: The magnitude of the cross product equals the area of the parallelogram formed by the two vectors.

Q5: What's the right-hand rule?
A: It's a mnemonic for determining the direction of the cross product vector - point fingers in direction of A, curl toward B, thumb points in direction of A×B.