3D Cross Product

The Cross Product

The cross product of $\mathbf{a} = (a_x, a_y, a_z)$ and $\mathbf{b} = (b_x, b_y, b_z)$ produces a new vector perpendicular to both:

$\mathbf{a} \times \mathbf{b} = (a_y b_z - a_z b_y,\; a_z b_x - a_x b_z,\; a_x b_y - a_y b_x)$

Key Properties

• Direction: perpendicular to both $\mathbf{a}$ and $\mathbf{b}$ (right-hand rule)
• Magnitude: $|\mathbf{a} \times \mathbf{b}| = |\mathbf{a}| \cdot |\mathbf{b}| \cdot \sin(\theta)$
• Anti-commutative: $\mathbf{a} \times \mathbf{b} = -(\mathbf{b} \times \mathbf{a})$
• If $\mathbf{a}$ and $\mathbf{b}$ are parallel: $\mathbf{a} \times \mathbf{b} = \mathbf{0}$

Geometric Meaning

The magnitude $|\mathbf{a} \times \mathbf{b}|$ equals the area of the parallelogram spanned by $\mathbf{a}$ and $\mathbf{b}$.

Memory Aid: The Determinant Formula

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

Expanding along the first row gives the formula above.

Applications

• Normal vectors to planes and surfaces
• Torque: $\boldsymbol{\tau} = \mathbf{r} \times \mathbf{F}$
• Area of triangles and parallelograms in 3D

Your Task

Implement void cross3(double ax, double ay, double az, double bx, double by, double bz, double *rx, double *ry, double *rz) that writes the cross product into the output pointers.