I am in a multivariable calculus course. Let a be the vector <-1 ,1, c> (with c any number) and b the vector <1, 1, 0>. The two vectors are orthogonal, as the dot product of a and b is 0. (The three positions of the vectors correspond with (i, j, k)).
I am trying to better understand the visualization of orthogonal vectors in three dimensions. Visually, why is it the case that the z-axis component of the vector a can be any number? I understand computationally that c times 0 will be zero. But I do not understand why the angle between a and b is $\frac\pi2$ regardless of the z-axis component of a. It makes sense to me visually that if c is 0, then <1,1,0> and <-1,1,0> would be orthogonal, but why does changing the z-axis component of a not change the angle between the vectors?
For reference, here is <1,1,0> and <-1, 1, 5>. How are these orthogonal?
