Cross Product Calculator Symbolab

Cross Product Formula:

\[ \mathbf{A} \times \mathbf{B} = (A_y B_z - A_z B_y,\; A_z B_x - A_x B_z,\; A_x B_y - A_y B_x) \]

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 standard cross product formula:

\[ \mathbf{A} \times \mathbf{B} = (A_y B_z - A_z B_y,\; A_z B_x - A_x B_z,\; A_x B_y - A_y B_x) \]

Where:

\( \mathbf{A} = (A_x, A_y, A_z) \) — First vector
\( \mathbf{B} = (B_x, B_y, B_z) \) — Second vector
\( \times \) — Cross product operator

Explanation: The cross product produces a vector perpendicular to both input vectors, with direction determined by the right-hand rule.

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 (area calculations).

4. Using the Calculator

Tips: Enter the x, y, and z components of 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. Dot product measures parallelism, cross product 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 cross product be used in 2D?
A: In 2D, the cross product is treated as a scalar (z-component of the 3D cross product with z=0).

Q4: How is the direction determined?
A: By the right-hand rule: point fingers in direction of A, curl toward B, thumb points in direction of A×B.

Q5: What's the geometric interpretation?
A: The magnitude equals the area of the parallelogram formed by the two vectors.