Waht exactly does it mean for 2 3D vectors to be orthogonal (in 3D) w.r.t dot product = 0

Question

I am suddenly puzzled by ho to know (when in 3D) with respect to which axis is the vector being rotated when the dot product between then is =0. for example: if i rotate 90degrees (pi/2 radians) along Z axis the vector [1 1 1] i get : [-1 1 1] which is at about 70° from [1 1 1] according to the formula $\cos \theta = \frac{x^\top y}{||x||\cdot||y||}$ I think i cannot visualize (anymore) what is happening when it comes to 3D ans its very frustrating... For me if i rotate along Z by 90 degrees the vectors should be at 90 degrees...? but again i must be missing something basic i guess?

Answer 1

You are measuring different angles. The 70 degrees you are getting is the angle between the two vectors. The 90 degrees is the angle between the projections of those vectors in the xy-plane.