Can someone please explain to me why the dot product of two perpendicular vectors are zero regardless of their magnitude ? I don't seem to understand how the magnitude of the vectors that generally have an effect on the dot product , is nullified if the vectors are perpendicular.
Can someone explain with an example and a visualization ?
On
By linearity of the dot product $$\lambda\vec u\cdot\mu\vec v=(\lambda\mu)(\vec u\cdot\vec v)$$
and $$\vec u\cdot\vec v=0\implies\lambda\vec u\cdot\mu\vec v=0.$$
so that the magnitude of orthogonal vectors does not matter.
Now the explanation of why a dot product can be zero depends on the definition that you use (Cartesian, geometric, trigonometric...). But if you admit that a dot product can be negative, then by continuity it can be zero as well.
Geometrically, one can think dot product as projection of one vector to other. Say, vector a is perpendicular to vector b, then vector a will have zero projection on vector b and vice versa.,(imagine shining a torch light from one to another).
Put a rod on floor, Case 1) rod is kept vertical, and shine a torch light kept at top of rod,shining towards floor, will you see any length of rod on floor(length of shadow of rod)? Regardless of length of rod, you will always see a point( which is 0 length).
Case 2) rod is inclined say θ with horizontal, you will now see some non zero length as shadow of rod, which is projection of rod on floor. It can be obtained from dot product calculation.