Cross Product Formula Calculator

Cross Product Formula:

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

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 cross product formula:

Where:

\( \vec{A} \) — First vector with components (A_x, A_y, A_z)
\( \vec{B} \) — Second vector with components (B_x, B_y, B_z)
\( \hat{i}, \hat{j}, \hat{k} \) — Unit vectors in x, y, z directions

Explanation: The cross product is calculated using the determinant of a 3×3 matrix composed of the unit vectors and the components of vectors A and B.

3. Importance of Cross Product

Details: The cross product is essential in physics and engineering for calculating torque, angular momentum, and determining perpendicular vectors. It's also used in computer graphics for calculating surface normals.

4. Using the Calculator

Tips: Enter all six components (x, y, z for both vectors A and B). The calculator will compute the resulting vector components.

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between dot product and cross product?
A: Dot product gives a scalar result, while cross product gives a vector result 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: The cross product is inherently a 3D operation, but you can extend 2D vectors to 3D by adding a z=0 component.

Q4: What's the right-hand rule?
A: It's 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: When is the cross product zero?
A: When the vectors are parallel or one of them is the zero vector.