Cross Product Calculator 2 Vectors

Cross Product Formula:

\[ \mathbf{A} \times \mathbf{B} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ A_x & A_y & A_z \\ B_x & B_y & B_z \end{vmatrix} = (A_yB_z - A_zB_y)\mathbf{i} - (A_xB_z - A_zB_x)\mathbf{j} + (A_xB_y - A_yB_x)\mathbf{k} \]

Cross Product (A × B):

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1. What is 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 Cross Product Works

The cross product of vectors A and B is calculated using 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: \( -(A_xB_z - A_zB_x) \)
Z-component: \( A_xB_y - A_yB_x \)

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

3. Applications of Cross Product

Uses: Cross products are essential in physics (torque, angular momentum), computer graphics (surface normals), engineering (moment of force), and many other fields involving 3D geometry.

4. Using the Calculator

Instructions: Enter the x, y, z components for both vectors A and B. The calculator will compute the cross product vector A × B.

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 that the vectors are parallel (or one/both are zero vectors).

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

Q4: Can cross product be used in 2D?
A: In 2D, the cross product is treated as a scalar (the z-component of what would be the 3D result).

Q5: What's the right-hand rule?
A: It's a mnemonic for determining the direction of the cross product vector - point fingers in A's direction, curl towards B, thumb points in A × B direction.