Cross Product Calculator Wolfram

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 the 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 that the vectors span.

2. How Does the Calculator Work?

The calculator uses the determinant formula for cross product:

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

Which expands to:

\( x \)-component: \( A_yB_z - A_zB_y \)
\( y \)-component: \( A_zB_x - A_xB_z \)
\( z \)-component: \( A_xB_y - A_yB_x \)

3. Importance of Cross Product

Applications: The cross product is essential in physics (torque, angular momentum), engineering (moment of force), computer graphics (surface normals), and many other fields involving 3D geometry.

4. Using the Calculator

Instructions: Enter the x, y, and 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 the magnitude of the cross product represent?
A: The magnitude 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 in 3D. In 2D, the cross product is a scalar, and in higher dimensions, more complex generalizations exist.

Q4: What's the right-hand rule?
A: A mnemonic for determining the direction of the cross product vector: point fingers in direction of A, curl towards B, thumb points in direction of A × B.

Q5: Can I calculate cross product for 2D vectors?
A: For 2D vectors (Aₓ, Aᵧ) and (Bₓ, Bᵧ), treat them as (Aₓ, Aᵧ, 0) and (Bₓ, Bᵧ, 0), which gives (0, 0, AₓBᵧ - AᵧBₓ).