Cross Product of Two Calculator

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 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:

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

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), computer graphics (surface normals), and engineering (moment of forces).

4. Using the Calculator

Instructions: Enter the x, y, and z components for both vectors A and B. The calculator will compute the resulting vector that is perpendicular to both input vectors.

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 quantity 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: Can cross product be calculated in 2D?
A: Technically no, but you can treat 2D vectors as 3D with z=0, resulting in a vector with only z-component.

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