12.4 Vectors and Cross Product

The cross product of two vectors $ extbf{a}$ and $ extbf{b}$, denoted $ extbf{a} \times \textbf{b}$, is a vector.
Defined for three-dimensional vectors only, contrasting with the dot product, which is a scalar.

Cross product can be calculated using determinants.
A determinant of order 3 can be expressed using second-order determinants.
The result can be calculated using the basis vectors $ extbf{i}, \textbf{j}, \textbf{k}$:
$\textbf{a} \times \textbf{b} = \begin{vmatrix} \textbf{i} & \textbf{j} & \textbf{k} \ a1 & a2 & a3 \ b1 & b2 & b3 \end{vmatrix}$ .

The vector $ extbf{a} \times \textbf{b}$ is orthogonal to both $ extbf{a}$ and $ extbf{b}$.
The direction is given by the right-hand rule.
Length of the cross product: $| \textbf{a} \times \textbf{b} | = |\textbf{a}| |\textbf{b}| \sin(\theta)$ , where $ heta$ is the angle between $ extbf{a}$ and $ extbf{b}$.
Vectors are parallel if and only if $\textbf{a} \times \textbf{b} = \textbf{0}$ .
The area of a parallelogram defined by $ extbf{a}$ and $ extbf{b}$ equals the magnitude of the cross product.

Scalar triple product $\textbf{a} \cdot (\textbf{b} \times \textbf{c})$ can be represented as a determinant:
$\begin{vmatrix} \textbf{a}1 & \textbf{a}2 & \textbf{a}3 \ \textbf{b}1 & \textbf{b}2 & \textbf{b}3 \ \textbf{c}1 & \textbf{c}2 & \textbf{c}_3 \end{vmatrix}$ .
The volume of the parallelepiped defined by vectors $ extbf{a}$, $ extbf{b}$, and $ extbf{c}$ is the absolute value of the scalar triple product.

Torque ($\textbf{\tau}$) is defined as \textbf{\tau} = \textbf{r} \times \textbf{F} , measuring the tendency to rotate about an axis.
The magnitude of torque is given by |\textbf{\tau}| = rF \sin(\theta) , where $ heta$ is the angle between position vector $ extbf{r}$ and force vector $ extbf{F}$.