3x3 Matrix Cross Product Calculator

3x3 Matrix Cross Product Formula:

\[ \text{CP} = \det\begin{pmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ V1_x & V1_y & V1_z \\ V2_x & V2_y & V2_z \end{pmatrix} \]

Vector 1 X-component:

Vector 1 Y-component:

Vector 1 Z-component:

Vector 2 X-component:

Vector 2 Y-component:

Vector 2 Z-component:

Cross Product Result:

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1. What is the 3x3 Matrix 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 calculated using the determinant of a 3x3 matrix with unit vectors i, j, k in the first row.

2. How Does the Calculator Work?

The calculator uses the cross product formula:

\[ \text{CP} = \det\begin{pmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ V1_x & V1_y & V1_z \\ V2_x & V2_y & V2_z \end{pmatrix} = (V1_y \cdot V2_z - V1_z \cdot V2_y)\mathbf{i} - (V1_x \cdot V2_z - V1_z \cdot V2_x)\mathbf{j} + (V1_x \cdot V2_y - V1_y \cdot V2_x)\mathbf{k} \]

Where:

\( V1 \) — First vector with components (V1_x, V1_y, V1_z)
\( V2 \) — Second vector with components (V2_x, V2_y, V2_z)
\( \mathbf{i}, \mathbf{j}, \mathbf{k} \) — Unit vectors along x, y, z axes

Explanation: The cross product produces a vector that is perpendicular to both input vectors, with magnitude equal to the area of the parallelogram they span.

3. Applications of Cross Product

Details: The cross product is essential in physics for calculating torque, in computer graphics for surface normals, in engineering for moment calculations, and in mathematics for determining perpendicular vectors.

4. Using the Calculator

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

5. Frequently Asked Questions (FAQ)

Q1: What is the geometric interpretation of the cross product?
A: The magnitude of the cross product equals the area of the parallelogram formed by the two vectors, and its direction is perpendicular to both vectors according to the right-hand rule.

Q2: How is the cross product different from the dot product?
A: The cross product results in a vector, while the dot product results in a scalar. The cross product measures perpendicularity, while the dot product measures parallel alignment.

Q3: What happens if the vectors are parallel?
A: If two vectors are parallel, their cross product is the zero vector (0, 0, 0).

Q4: Can the cross product be calculated in 2D?
A: The standard cross product is defined only in 3D space. For 2D vectors, we typically consider them as 3D vectors with z=0, resulting in a cross product with only a z-component.

Q5: What is the right-hand rule?
A: Point your index finger in the direction of the first vector, your middle finger in the direction of the second vector, and your thumb will point in the direction of the cross product.