Vector Cross Product Calculation

Cross Product Formula:

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

Vector A X-component:

Vector A Y-component:

Vector A Z-component:

Vector B X-component:

Vector B Y-component:

Vector B Z-component:

Cross Product Result (Vector C):

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1. What Is The Vector 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 has important applications in physics, engineering, and computer graphics.

2. How Does The Calculator Work?

The calculator uses the cross product formula:

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

Where:

\( \vec{A} = (A_x, A_y, A_z) \) — First input vector
\( \vec{B} = (B_x, B_y, B_z) \) — Second input vector
\( \vec{C} = (C_x, C_y, C_z) \) — Resulting cross product vector

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

3. Applications Of Cross Product

Details: The cross product is used in calculating torque, determining surface normals in 3D graphics, finding the moment of a force, and in electromagnetic theory to calculate the Lorentz force.

4. Using The Calculator

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

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 the plane containing both vectors.

Q2: Is the cross product commutative?
A: No, the cross product is anti-commutative: A × B = - (B × A).

Q3: What is the right-hand rule?
A: The right-hand rule determines the direction of the cross product: point fingers in direction of A, curl toward B, and thumb points in direction of A × B.

Q4: When is the cross product zero?
A: The cross product is zero when the vectors are parallel or when at least one vector is a zero vector.

Q5: Can cross product be defined in 2D space?
A: The cross product is specifically defined for 3D space. In 2D, the analogous operation returns a scalar representing the signed area of the parallelogram.