Cross Product Calculator Formula For Two

Cross Product Formula:

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

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 spanned by the two vectors.

2. How Does the Calculator Work?

The calculator uses the standard cross product formula:

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 equal to the area of the parallelogram they span.

3. Applications of Cross Product

Details: The cross product is used in physics (torque, angular momentum), computer graphics (surface normals), engineering (moment of force), and mathematics (determining orthogonality).

4. Using the Calculator

Tips: Enter all six components (x,y,z for both vectors). The calculator will compute the resulting vector components along i, j, and k directions.

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between dot product and cross product?
A: Dot product gives a scalar quantity and measures projection, while cross product gives a vector quantity and measures 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: Can you compute cross product in 2D?
A: In 2D, the cross product is conceptually treated as the z-component of the 3D cross product with z=0.

Q4: What's the right-hand rule?
A: It's a mnemonic for determining the direction of the cross product vector: point fingers in A's direction, curl towards B, thumb points in A×B direction.

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