Solve Rotation Matrix Problem 1.9 of DJGriffiths - (1,1,1) Direction

[Living_Dog]

Problem 1.9 of DJGriffiths asks for the rotation matrix about the (1,1,1) direction.

I thought I could rotate about z 45 degrees (R': x -> x'), then rotate about y' (R'': x' -> x''). How do I combine the two rotations to determine the final single rotation matrix... R = R''*R' or R = R'*R'' ?

thx in advance!
-LD

EDIT: after working on this some more, I realized that this is not the same rotation. The rotation is not from x->x'->x''. Rather it is a rotation around the (1,1,1) direction.

Having checked what few texts I have I still do not know how to make the rotation matrix.

After some thought ...wouldn't x go to the z position to satisfy this rotation? I'm thinking that if you look down the (1,1,1) line toward the origin, then the 3 axes make a 120^o angle with each other. Thus the x-axis will spin into the original z-axis position. Also, z->y and y->x.

tia!
-LD

EDIT: after working on this some more, I found the solution. Perhaps it is a bit pragmatic since I used the above fact. So if anyone knows how to do this in general - for any direction - then I would like the answer. (I think I need a book on crystallographic rotations - yikes!)

100 -> 001
010 -> 100
001 -> 010

Now apply these conditions to the equation: A' = R*A. Then each equation will result in the immediate solution for 3 of the rotation matrix elements.

(0 0 1) = R (1 0 0) yields:
$$R_{11} = 0$$
$$R_{21} = 0$$
$$R_{31} = 1$$

(1 0 0) = R (0 1 0) yields:
$$R_{12} = 1$$
$$R_{22} = 0$$
$$R_{32} = 0$$

(0 1 0) = R (0 0 1) yields:
$$R_{13} = 0$$
$$R_{23} = 0$$
$$R_{33} = 1$$

Thus, the R matrix is:

$$\left( \begin{array}{ccc} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0 \end{array} \right)$$

...sorry I asked this and then found the solution... but as I said above, this is very pragmatic and I would really like to know how to do this for any angle of rotation about any direction in general.

Thanks!
-LD

 

[AKG]

Let R(a) be the rotation matrix about the z-axis through angle a. Let R(v) be the rotation matrix that sends the vector v to the z axis. Then the rotation about v through angle a is:

R(v)^-1R(a)R(v)

You know how to find R(a). To find R(v), first find the polar and azimuthal angles of v/|v|, suppose they are b and c respectively. Then R(v) is a rotation about the z axis through the angle -c, followed by a rotation about the y-axis through angle -b.

 

[Living_Dog]

Let R(a) be the rotation matrix about the z-axis through angle a. Let R(v) be the rotation matrix that sends the vector v to the z axis. Then the rotation about v through angle a is:

R(v)^-1R(a)R(v)

You know how to find R(a). To find R(v), first find the polar and azimuthal angles of v/|v|, suppose they are b and c respectively. Then R(v) is a rotation about the z axis through the angle -c, followed by a rotation about the y-axis through angle -b.

Thanks!

BTW, isn't that something called a similarity transformation? (I seem to remember it from Goldstein's chapter on Euler angles... so much forgotten down the corners of my mind :/ )