TITLE: which vectors are perpendicular to each other?
QUESTION [5 upvotes]: which vectors are perpendicular to each other?
$\vec a = (1, -2, 3)$, $\vec b = (5, 4, 1)$, $\vec c = (1, 0, -5)$
Do i just take the dot product of 2 of them. If the dot product they are at $90^\circ$? But how do i know if there perpendicular?

REPLY [5 votes]: This is a numerical answer since explanatory answers have already been given.
$\vec a = (1,-2,3)$ and $\vec b=(5,4,1)$ are perpendicular (orthogonal)
$\vec b = (5,4,1)$ and $\vec c=(1,0,-5)$ are also perpendicular.
Why? 
$$\vec a\cdot \vec b=\begin{bmatrix}1\\-2\\3\end{bmatrix}\cdot\begin{bmatrix}5\\4\\1\end{bmatrix}=1\cdot 5-2\cdot 4+3\cdot 1=5-8+3=0$$
$$\vec b\cdot \vec c=\begin{bmatrix}5\\4\\1\end{bmatrix}\cdot\begin{bmatrix}1\\0\\-5\end{bmatrix}=5\cdot 1+4\cdot 0+1\cdot -5=5-0-5=0$$
This is because their scalar products are both zero.