0
$\begingroup$

I'm having trouble with the following problem:

Rotated x’ axes are chosen making angles with the x axes as shown in the following table:

$$\begin{array}{ccc}&x'&y'&z'\\x&90^{\circ}&45^{\circ}&135^{\circ}\\y&45^{\circ}&60^{\circ}&60^{\circ}\\z&45^{\circ}&120^{\circ}&120^{\circ}\end{array}$$

Calculate the rotation matrix.

I am new to rotation matrices, but have been reading the wiki and Wolfram pages. From what I understand, if I can obtain the yaw, pitch, and roll angles from the old axes to produce the new axes, then I can easily obtain the rotation matrix. However, I do not understand how to use the provided angles in order to get the yaw, pitch, and roll.

Is this understanding correct? And if so, how can I get those 3 angles? Thanks.

$\endgroup$
2
$\begingroup$

Let $e_x=(1,0,0),\,e_y=(0,1,0)$ and $e_z=(0,0,1)$ be unit vectors pointing along the $x$, $y$ and $z$ axes. Let $R$ be the desired rotation matrix and $u_x=Re_x,\,u_y=Re_y,\,u_z=Re_z$. These are vectors pointing along the $x'$, $y'$ and $z'$ axes, and they are unit vectors since $R$ is a rotation. Each angle you are given gives information about a dot product between these vectors; for example $$ e_x\cdot u_x=\cos 90^\circ=0 $$ (since $e_x$ and $u_x$ are unit vectors). Note that $e_x\cdot u_x$ is also the first component of $u_x$. Using this you can determine the vectors $u_x,\,u_y$ and $u_z$. But these vectors are also the columns of $R$, so you can then put them side by side to construct $R$.

| cite | improve this answer | |
$\endgroup$
  • $\begingroup$ I see! Thanks! Your answer is helping me understand the meaning of rotation matrices much more clearly. I feel that I had lost sight of the geometric picture previously. $\endgroup$ – spinodal Sep 6 '16 at 3:39

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.