Rotation of axes give the same point in space?

Question

I am playing around with rotation of axes formulas and not getting it. I don't understand how this rotates anything when it is just giving you different coordinates for the SAME point in space. How can anything be rotated when it's giving the same points each time?

Answer 1

In Figure 1 of http://www.stewartcalculus.com/data/CALCULUS%20Early%20Vectors/upfiles/RotationofAxes.pdf we see that indeed the point P doesn't move (at least with respect to the page on which the figure is drawn). But while the $y$-axis goes straight up and down on the page, the $Y$-axis is leaning to the left; and while the $x$-axis runs directly right and left in a level line on the page, the $X$-axis is tilted so it is higher on the right and lower on the left.

Suppose you had a sheet of transparent plastic with a rectangular grid (like graph paper) printed on it. You could align this grid with the $x$ and $y$ axes (so two of the lines on the grid lie exactly on those two axes) by laying the transparent sheet carefully on the page where the figure is drawn. But if you later want to align that grid with the $X$ and $Y$ axes instead, you will have to rotate the transparent sheet somehow in order to make the lines of the grid lie on those new axes.

In short, the figure is illustrating a rotation of the axes. Instead of the original axes (and their associated grid), we use a new set of axes oriented in a different direction. Things that are not the axes (such as point P, whose position is being measured relative to the axes) don't move.