Cross Product Calculator With Variables

Cross Product Formula:

\[ \vec{a} \times \vec{b} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ a_x & a_y & a_z \\ b_x & b_y & b_z \\ \end{vmatrix} = (a_yb_z - a_zb_y)\mathbf{i} - (a_xb_z - a_zb_x)\mathbf{j} + (a_xb_y - a_yb_x)\mathbf{k} \]

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 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 is calculated using the determinant formula:

\[ \vec{a} \times \vec{b} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ a_x & a_y & a_z \\ b_x & b_y & b_z \\ \end{vmatrix} \]

Which expands to: \[ (a_yb_z - a_zb_y)\mathbf{i} - (a_xb_z - a_zb_x)\mathbf{j} + (a_xb_y - a_yb_x)\mathbf{k} \]

Properties:

Anti-commutative: a × b = - (b × a)
Distributive over addition
Compatible with scalar multiplication

3. Applications of Cross Product

Common Uses: Calculating torque, finding normal vectors, determining if vectors are parallel, and in computer graphics for lighting calculations.

4. Using the Calculator

Instructions: Enter all six components (x,y,z for both vectors). The calculator will compute the resulting vector's components.

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between dot product and cross product?
A: Dot product gives a scalar, cross product gives a vector perpendicular to both input vectors.

Q2: Can you compute cross product in 2D?
A: Not directly - it's inherently a 3D operation. For 2D vectors, you'd add z=0.

Q3: What does a zero cross product mean?
A: It means the vectors are parallel (or at least one is zero).

Q4: How is cross product related to area?
A: The magnitude of the cross product equals the area of the parallelogram formed by the two vectors.

Q5: Why is cross product anti-commutative?
A: Because the right-hand rule determines direction, and swapping vectors reverses this direction.