Cross Product Calculator Symbolab

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 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: \( A_xB_y - A_yB_x \)

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

3. Applications of Cross Product

Details: The cross product is used in physics (torque, angular momentum), computer graphics (surface normals), engineering (moment of forces), and mathematics (calculating areas and volumes).

4. Using the Calculator

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

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 you calculate cross product in 2D?
A: Technically no, but you can extend 2D vectors to 3D by adding a z-component of 0, resulting in a vector with only a 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).