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Given the vector $x=[1, 3, -7]^T$ in the basis $[1, 0, 0]^T$, $[0, 1, 0]^T$, and $[0, 0, 1]^T$ (Cartesian coordinates) perform the following operations:

Rotate 45 degrees about the x-axis then rotate 30 degrees about the y-axis then rotate -10 degrees about the z-axis

I have absolutely no idea how to start this problem. I can't find any examples. I will continue to look on this website for examples as well.

I do know that the rotation matrix is:

$$ Q= \left[ \begin{array}{cc} \cos(\theta) & -\sin(\theta) \\ \sin(\theta) & \cos(\theta) \end{array} \right] $$

The question does go on to ask for reflections and projections about and onto various axis.

Any help would be greatly appreciated. All of you math wizards have given me some great help!! Thanks!!

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1 Answer

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The matrix you need will be $3 \times 3$ and it is formed by taking the matrix you wrote and adjusting it a bit. Basically, you want to fix the axis of the rotation and twist the other two directions. To fix the $z$-direction use: $$ R = \left[ \begin{array}{ccc} \cos(\theta) & -\sin(\theta) & 0 \\ \sin(\theta) & \cos(\theta) & 0 \\ 0 & 0 & 1 \end{array} \right] $$ To fix the $y$-direction (which means $x$ and $z$ directions are rotated) use: $$ R = \left[ \begin{array}{ccc} \cos(\theta) & 0 & -\sin(\theta) \\ 0 & 1 & 0 \\ \sin(\theta) & 0 & \cos(\theta) \end{array} \right] $$ and finally, to rotate around the $x$-axis: $$ R = \left[ \begin{array}{ccc} 1 & 0 & 0 \\ 0 & \cos(\theta) & -\sin(\theta) \\ 0 & \sin(\theta) & \cos(\theta) \end{array} \right] $$ maybe you should try some simple examples with my suggestions to see how these work. Pick an easy angle and a simple vector and test the transformations.

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