Vector Cross Multiplication Calculator

Vector Cross Product Formula:

\[ \mathbf{u} \times \mathbf{v} = (u_y v_z - u_z v_y,\; u_z v_x - u_x v_z,\; u_x v_y - u_y v_x) \]

Vector u (x component):

Vector u (y component):

Vector u (z component):

Vector v (x component):

Vector v (y component):

Vector v (z component):

Cross Product Result (u × v):

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1. What is Vector 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 they span.

2. How Does the Calculator Work?

The calculator uses the standard cross product formula:

\[ \mathbf{u} \times \mathbf{v} = (u_y v_z - u_z v_y,\; u_z v_x - u_x v_z,\; u_x v_y - u_y v_x) \]

Where:

\( \mathbf{u} = (u_x, u_y, u_z) \) — First vector
\( \mathbf{v} = (v_x, v_y, v_z) \) — Second vector
\( \times \) — Cross product operator

Explanation: Each component of the resulting vector is calculated from the other two components of the input vectors.

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 cross product vector which is 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, while cross product gives a vector quantity perpendicular to both input vectors.

Q2: What does the magnitude of the cross product represent?
A: The magnitude equals the area of the parallelogram formed by the two vectors.

Q3: Can you compute cross product in 2D?
A: In 2D, the cross product is treated as a scalar (z-component of 3D cross product with z=0).

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

Q5: What happens if vectors are parallel?
A: The cross product is the zero vector (0, 0, 0) when vectors are parallel or antiparallel.