how to calculate the angle in the $x-y, y-z, x-z$ plane given only $3D$ vector direction and magnitude?

Question

Please help me solve this. I have been thinking of all sorts of ways to solve this but can't figure out how :(. OK here's the problem: I am given a three dimensional velocity vector (I know the magnitude of this vector and I know what angle this vector makes w.r.t. one of the axis, say the $x$-axis.) What I want to determine is what angle does this three-dimensional vector make with $X-Y$, $Y-Z$ and $X-Z$ plane. Another way to look at this is if we project this three-dimensional vector in the $X-Y$ plane what is the angle between this vector and the $x$- axis (or the $y$- axis)? I do not know the velocity components of this vector in the $x$, $y$ or $z$-axis. As a matter of fact, these $x$, $y$ and $z$-velocity components are what I aim to calculate from determining the angle the $3D$ vector makes with each plane.

Answer 1

You need three scalar quantities to specify a vector in a three-dimensional space. If you know only the magnitude of the vector and the angle it forms with one axis, those two scalar quantities are not enough to fully determine the vector -- it could be any vector with the given magnitude on a cone around that axis. You need one more quantity to fix the vector; then you can calculate the angles and components you're looking for.

P.S.: Note that the title and body of the question are inconsistent. Specifying a direction in three-dimensional space requires two scalar quantities; since you only have one angle, you don't know the direction of the vector.