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

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 3-D vector makes with each plane.

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Maybe you want direction cosines: en.wikipedia.org/wiki/Direction_cosine – Ross Millikan Jul 28 '11 at 23:19
Wow this post has been a big help. If I understand direction cosines right, it is similar to imagining a cone whose dimensions are given by 3-D vector and angle wrt x-axis say (lets call this vector R and angle phi). Now, if we imagine the cone completing a revolution around the x-y plane, and we ask ourselves at what angle wrt x-axis does the cone cut through the x-y plane -- it is the same angle as phi. – Rhea Jul 29 '11 at 7:10
..also, the length of the cone does not change as it cuts the x-y plane-- so this reduces the problem to a 2D problem in the x-y plane with both magnitude and angle wrt to x-axis specified. Then using simple trig (R cos phi) we can find x-component of this 3-D vector R. – Rhea Jul 29 '11 at 7:10
Am I on the right track on this or not? – Rhea Jul 29 '11 at 23:02
you are right that given a vector length and angle with the x axis, you have a 2D problem. There will be two vectors that satisfy that: $(v \cos \theta, v \sin \theta)$ and $(v \cos \theta, -v \sin \theta)$ – Ross Millikan Jul 30 '11 at 17:06