Out of context when you see two linear functions like:
$$y_{2} = 16x$$ and $$y_{2}= 16x + 3$$
something that probably easily comes to your mind is that they are parallel. It is just effortless to notice. Pretty much same thing for perpendicular case, one can easily observe if two linear functions are perpendicular to each other without doing any calculations.
Two vectors are parallel if they are "multiple" of each other (if I understand that correctly).
So for example:
$\vec{a} = [1, 2, 3] \text{ and } \vec{b} = [111, 222, 333]$
are parallel. This is something that can be often noticed without doing calculations.
Is there a similar trick for quick-telling if two vectors are perpendicular to each other, just by a brief glance?
Best Answer
There are quick tricks for certain forms, however they certainly do not imply all perpendicular vectors follow these patterns. Therefore I would consider my following discussion useful for coming up with perpendicular vectors, not necessarily for showing if a vector is perpendicular. As it is best to compute ur defined inner product, dot product in this case, and seeing if it is equal to zero.
ex.1) For the simple two dimensional case. $$v=(x,y)$$ $$if\\w=(-y,x)$$ this implies that their dot product, defined as the component wise multiplication of the two vectors and then taking their sum is the following: $$(x)(-y)+(y)(x)=(0,0)$$ hence by definition perpendicular However, this statement once again I'll remind you does not go both ways. As there are two dimensional vectors that are perpendicular but do not hold this relationship Say, $$(3,4)*(-8,6)=(0,0)$$ However, one can notice the familiar pattern: $$if\\ v=(x,y)$$ then $w$ perpendicular to $v$ if$$\\w=(-2y,2x)$$ and this indeed works for any arbitrary coefficient, not just $2$.
ex.)2)$$(x,y,z)*(-y,x,0)=(0,0,0)$$ also works once again with any multiples
ex.)3.) $$(w,x,y,z)*(-x,w,-z,y)$$ $$(w,x,y,z)*(-z,-y,x,w)$$
also both work once again with any multiples
The pattern can be summarized as switching two places of the vectors components and making one negative, hence it is more useful with even dimensions. But by placing zeroes can also use with odd dimensions