A Transformation Matrix question

[Saladsamurai]

Homework Statement

Find the transformation matrix 'R' that describes a rotation by 120 degrees about an axis from the origin through the point (1,1,1). The rotation is clockwise as you look down the axis toward the origin.

Homework Equations

<br /> \left( \begin{array}{c} A&#039;_x \\ A&#039;_y \\ A&#039;_z \end{array} \right) = <br /> <br /> \left( \begin{array}{ccc} <br /> R_{xx} &amp; R_{xy} &amp; R_{xz} \\ <br /> R_{yx} &amp; R_{yy} &amp; R_{yz} \\<br /> R_{zx} &amp; R_{zy} &amp; R_{zz}<br /> \end{array} \right)<br /> <br /> \left( \begin{array}{c} A_x \\ A_y \\ A_z \end{array} \right)<br /> <br /> <br /> <br />

The Attempt at a Solution

I am just very confused by the wording of the question. I am used to talking about the transformation of a vector... I am not sure what is being transformed here... the coordinate system?

Here is the solution:

I am having a hard time deciphering the problem statement even looking at the solution. It is clear that he wanted us to "swap" axes. But I am not sure exactly what is happening here.

 

[lanedance]

you can imagine it a few different ways, i would think of it as follows:
- The global coordinate frame does not change.
- The operation is a transformation of a vector within that frame (it maps each vector to a vector).

so imagine the line from the origin to (1,1,1)

Now start with the vector (1,0,0), this will be transformed to (0,0,1).

The global co-ordinate frame doesn't change, but a vector on the x-axis is mapped to a vector on the z axis.

By considering the action on each of the basis vectors it shoudl be pretty startightforward to write down the matrix.