Vector Calculus

Cross Product

Calculating the Cross Product

Theorems and Properties

• $\mathbf{x}\times \mathbf{y}$ is orthogonal to both $\mathbf{x}$ and $\mathbf{y}$.
• If given $\theta \mid (0\le \theta \le \pi )$ between $\mathbf{x}$ and $\mathbf{y}$, then $|\mathbf{x}\times \mathbf{y}|=|\mathbf{x}||\mathbf{y}|sin(\theta )$.
• If $\mathbf{x}$ and $\mathbf{y}$ are non-zero and $\mathbf{x}\times \mathbf{y}=0$, then $\mathbf{x}$ and $\mathbf{y}$ are parallel.

Area of a Parallelogram

$A=|\mathbf{x}\times \mathbf{y}|=|\mathbf{x}||\mathbf{y}|sin(\theta )$

Scalar Triple Product

$\mathbf{x}\cdot (\mathbf{y}\times \mathbf{z})$

Parallelepiped Volume

$$V=Ah=|\mathbf{y}\times \mathbf{z}||\mathbf{x}||cos\theta |=|\mathbf{x}\cdot (\mathbf{y}\times \mathbf{z})|$$

Therefore,

$$V=|\mathbf{x}\cdot (\mathbf{y}\times \mathbf{z})|$$

Note: If $V=0$, then $\mathbf{x},\mathbf{y},\mathbf{z}$ are coplanar.