Vector Calculus

Cross Product

Calculating the Cross Product

Theorems and Properties

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

Assume $\vec{x}$, $\vec{y}$, and $\vec{z}$ are vectors and $c$ is a scalar,

1. $\vec{x}\times \vec{y}=-\vec{y}\times \vec{x}$
2. $(c\vec{x})\times \vec{y}=c(\vec{x}\times \vec{y})=\vec{x}\times (c\vec{y})$
3. $\vec{x}\times (\vec{y}+\vec{z})=\vec{x}\times \vec{y}+\vec{x}\times \vec{z}$
4. $(\vec{x}+\vec{y})\times \vec{z}=\vec{x}\times \vec{z}+\vec{y}\times \vec{z}$
5. $\vec{x}\cdot (\vec{y}\times \vec{z})=(\vec{x}\times \vec{y})\cdot \vec{z}$
6. $\vec{x}\times (\vec{y}\times \vec{z})=(\vec{x}\cdot \vec{z})\vec{y}-(\vec{x}\cdot \vec{y})\vec{z}$

Area of a Parallelogram

$A=|\vec{x}\times \vec{y}|=|\vec{x}||\vec{y}|sin\theta$

Scalar Triple Product

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

Parallelepiped Volume

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

Therefore,

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

Note: If $V=0$, $\vec{x},\vec{y},\vec{z}$ all lie on the same plane.