I have two planes and I wanna check whether they are orthogonal and I am not sure if I am correct.
If I understand this correctly an orthogonality is just a multi-vector extension of perpendicularity so checking whether two planes are orthogonal is the same as checking whether they are perpendicular. If I have two planes:
\begin{equation}
2a − b + c = 6
\end{equation}
\begin{equation}
a + 5b + 3c = 2
\end{equation}
I just need to check their normal vector dot product which is:
\begin{equation}
v_1 \cdot v_2 = <2, -1,1> \cdot < 1, 5, 3> = 2 – 5 + 3 = 0
\end{equation}
and since its 0 the two planes are orthogonal right? Or is this just perpendicularity and to check whether they are orthogonal I need to do something more
Best Answer
They’re not quite the same in my mind. I think of orthogonality as a generalization of the geometric idea of perpendicularity. One can speak of a pair of functions being orthogonal even though there’s no obvious geometric interpretation of this relationship. When you’re working in a Euclidean space they’re pretty much synonymous, although even there “orthogonal” is the more general term: the zero vector is orthogonal to everything, but one doesn’t usually describe it as being perpendicular to anything.
For the curious, the word “perpendicular” derives from the Latin for “plumb line,” so at root it has the notion of hanging straight down or standing straight up from the ground. The word “orthogonal,” on the other hand, means “right angled” or “straight angled,” so even in their etymology the latter seems to me like a more broadly-applicable term.