Cross Vector Calculator

Cross Product Formula:

\[ A × B = (A_y B_z - A_z B_y, A_z B_x - A_x B_z, A_x B_y - A_y B_x) \]

Vector A (x):

Vector A (y):

Vector A (z):

Vector B (x):

Vector B (y):

Vector B (z):

Cross Product Result (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 cross product formula:

\[ A × B = (A_y B_z - A_z B_y, A_z B_x - A_x B_z, A_x B_y - A_y B_x) \]

Where:

\( A \) — First vector (Aₓ, Aᵧ, A_z)
\( B \) — Second vector (Bₓ, Bᵧ, B_z)
× — Cross product operator

Explanation: The result is a new vector perpendicular to both A and B, following the right-hand rule.

3. Importance of Cross Product

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

4. Using the Calculator

Tips: Enter all six components (x,y,z for 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, while cross product gives a vector perpendicular to both input vectors.

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

Q3: Why is the cross product only defined in 3D?
A: The perpendicular vector concept only works consistently in three dimensions, though 7D has a similar operation.

Q4: What's the right-hand rule?
A: Point fingers in direction of A, curl toward B; thumb points in direction of A × B.

Q5: What if the cross product is zero?
A: The vectors are parallel (or one is zero).