I am answering the following question from Pearson Edexcel AS and A level Further Mathematics, Book 1/AS.

I drew this out (below), noting x, y, z basis are transformed to coordinates of columns 1, 2, 3 in matrix M. The theta is 30 degrees, but x and z are also flipped. The rotation is in the plane y=0 (red), x axis (blue) and z axis (orange) mapped to those (rough in image) positions. To me, this is a 150 degree rotation anti-clockwise. However, the result is 210 degree anti-clockwise rotation about the y-plane. I don't understand how this is possible, as if it was 210 degree rotation the x-basis should be negative in the Z component.

Thanks

## Best Answer

You are looking at the $x,z$ plane (the plane $y=0$) "from underneath." In your figure, the positive $y$ axis is on the side of the plane farther from the person viewing the figure, and the negative $y$ axis is on the side of the plane closer to the person viewing the figure.

If you were talking about the direction of a rotation in the $x,y$ plane (where $z=0$) you would look down upon the plane from a position on or near the positive $z$ axis in order to decide whether a rotation is clockwise or anti-clockwise. That way the answer agrees with the answer for a 2D rotation.

In general, the direction of a rotation (clockwise or anti-clockwise) is going to depend upon the point of view from which you look at the rotation. Try drawing the figure so that the positive $y$ axis is in front of the plane $y=0$ from your point of view, rather than behind the plane. That is the point of view that is most likely desired for this question.

One way to make such a drawing is to use the line on the lower left as the positive $x$ axis and the line on the lower right as the positive $y$ axis. Then you may see that the rotations you have drawn are $150$ degrees clockwise (viewed from that direction), which is the same result as a rotation $210$ degrees anti-clockwise.