# Find vector components from magnitude and angles

I am aware of this Get vector components from from magnitude and angle and Finding Vectors from their angle and magnitude but they do not really solve my question.

If I have theta = 20 degrees and I am given a magnitude of m = 5 I can easily figure out the x and y components using m * cos(theta) = x and m * sin(theta) = y. However, I am not sure how this expands into 3 dimensions. Or, in other words, how would I find the z component? In the 3 dimensional case I also have access to two more angles. Theta being the angle about the z-axis, phi being the angle about the x-axis, and gamma being the angle about the y-axis.

Therefore the final question is. Given angles about the xyz axes, and a magnitude of a vector, how do I get the components x, y, and z of that vector?

• what is $\theta$ exactly? If you have only the angle to one axes the solution is not unique. Oct 3, 2022 at 16:38
• In the two-dimensional case the angle isn't an angle about either the $x$ axis or the $y$ axis. It's an angle counterclockwise from the $x$ axis. You can have angles from all three axes in 3D, but the "counterclockwise" part doesn't make sense any more so the angles only go from $0$ to $\pi,$ and you get all the coordinates of the vector with cosines. If you actually want angles about axes then two axes are enough, since the direction of a vector has only two degrees of freedom in 3D. Oct 3, 2022 at 16:40
• ($m \cos \phi$, $m \cos \gamma$, $m \cos \theta$) : does this work? Oct 3, 2022 at 16:42