Calculate Vector Cross Product

Vector Cross Product Formula:

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

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 (A × B):

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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 determinant formula:

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

Which expands to:

X component: \( A_yB_z - A_zB_y \)
Y component: \( A_zB_x - A_xB_z \)
Z component: \( A_xB_y - A_yB_x \)

Properties: The resulting vector is orthogonal to both input vectors, and its magnitude equals the area of the parallelogram formed by A and B.

3. Applications of Cross Product

Details: Cross products are essential in physics (torque, angular momentum), computer graphics (surface normals), and engineering (moment of forces).

4. Using the Calculator

Tips: Enter all six components (x,y,z for both vectors). The calculator will compute the cross product 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 a zero cross product mean?
A: A zero cross product indicates the vectors are parallel (or at least one is zero).

Q3: Is cross product commutative?
A: No, A × B = - (B × A). It's anti-commutative.

Q4: Can you compute cross product in 2D?
A: The 2D equivalent would be a scalar representing the signed area of the parallelogram.

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