Cross or vector product | Improve Society

The cross or vector product of $\bold{A}$ and $\bold{B}$ is a vector $\bold{C} = \bold{A} \times \bold{B}$ (read $\bold{A}$ cross $\bold{B}$ ). The magnitude of $\bold{A}\times\bold{B}$ is defined as the product of the magnitudes of $\bold{A}$ and $\bold{B}$ and the sine of the angle between them. The direction of the vector $\bold{C} = \bold{A}\times\bold{B}$ is perpendicular to the plane of $\bold{A}$ and $\bold{B}$ and such that $\bold{A}$ , $\bold{B}$ and $\bold{C}$ form a right-handed system. In symbols,
$\bold{A}\times\bold{B} = AB\sin{\theta}\bold{u},\ 0\le\theta\le\pi\cdots(5)$
where $\bold{u}$ is a unit vector indicating the direction of $\bold{A}\times\bold{B}$ . If $\bold{A} = \bold{B}$ or if $\bold{A}$ is parallel to $\bold{B}$ , then $\sin\theta = 0$ and we define $\bold{A}\times\bold{B} = 0$ .

//en.wikipedia.org/wiki/Vector_product

The following laws are valid:

$\bold{A} \times \bold{B} = - \bold{B} \times \bold{A}$

$\bold{A} \times (\bold{B} + \bold{C}) = \bold{A}\times\bold{B} + \bold{A} \times \bold{C}$

$m( \bold{A} \times \bold{B}) = (m \bold{A}) \times \bold{B} = \bold{A} \times (m \bold{B}) = (\bold{A} \times \bold{B})m$

$\bold{i} \times \bold{i} = \bold{j} \times \bold{j} = \bold{k} \times \bold{k} = 0,\ \bold{i} \times \bold{j} = \bold{k},\ \bold{j} \times \bold{k} = \bold{i},\ \bold{k} \times \bold{i} = \bold{j}$

If $\bold{A} = A_1\bold{i} + A_2\bold{j} + A_3\bold{k}$ and $\bold{B} = B_1\bold{i} + B_2\bold{j} + B_3\bold{k}$ , then
$\displaystyle \bold{A} \times \bold{B} = \left|\begin{array}{ccc} \bold{i} & \bold{j} & \bold{k} \\ A_1 & A_2 & A_3 \\ B_1 & B_2 & B_3 \end{array}\right| = \left|\begin{array}{cc} A_2 & A_3 \\ B_2 & B_3 \end{array}\right| \bold{i} - \left|\begin{array}{cc} A_1 & A_3 \\ B_1 & B_3 \end{array}\right| \bold{j} + \left|\begin{array}{cc} A_1 & A_2 \\ B_1 & B_2 \end{array}\right| \bold{k}$
$|\bold{A} \times \bold{B}|$ is the area of a parallelogram with sides $\bold{A}$ and $\bold{B}$ .

If $\bold{A} \times \bold{B} = 0$ and $\bold{A}$ and $\bold{B}$ are not null vectors, then $\bold{A}$ and $\bold{B}$ are parallel.

Note that communicative law for cross products is failed.