3.2 Vector multiplication reminder

It is presumed that you are familiar with vector multiplication.

Here are some questions to ask yourself as a refresher:

$\vec{A}\cdot \vec{B}$ . Is it a vector or a scalar?

Answer: scalar $$\vec{A}\cdot \vec{B}=A_x{B}_x+A_y{B}_y+A_z{B}_z$$

$\vec{A}\times \vec{B}$ . Is it a vector or a scalar?

Answer: Vector $${\displaystyle \begin{array}{c}{(\vec{A}\times \vec{B})}_x=A_y{B}_z-A_z{B}_y\\ {(\vec{A}\times \vec{B})}_y=A_z{B}_x-A_x{B}_z\\ {(\vec{A}\times \vec{B})}_z=A_x{B}_y-A_y{B}_x\end{array}}$$

What is $\vec{A}\times \vec{A}$

Answer: 0

What is $\vec{A}\cdot (\vec{A}\times \vec{B})$

Answer: 0

Another useful thing to remember $$\vec{A}\cdot (\vec{B}\times \vec{C})=(\vec{A}\times \vec{B})\cdot \vec{C}$$ $$=(\vec{C}\times \vec{A})\cdot \vec{B}$$ This is the scalar triple product which is the volume of a parallelopiped whoseedges are given by $\vec{A},\vec{B},\vec{C}$ .