There are a number of ingredients that go into the general formula for rotation in the (x,y) plane. Here is a set of practice exercises to work and some explanations for you to contemplate. They may help you put the whole picture together.

When points A, B, C are on a line, the ratio AC/AB is taken to be a signed ratio, which is negative is A is between B and C.

1. Draw on graph paper the point P with coordinates (3,4). Then P' is obtained by rotating P by 90 degrees with center O = (0,0). Draw P' on your graph paper. What are the coordinates of P'?
2. If a point Q has coordinates (a,b). If Q' is obtained by rotating Q by 90 degrees with center O = (0,0), what are the coordinates of Q?
3. Check that this works even if Q is not in the first quadrant, for example, if Q1=(-3,2), what is Q1'?
4. Or if Q2 = (-3,-2), what is Q2'?

These figures may help you figure out or explain the formula.

Suppose that on the plane, O = the origin = (0,0), E1 = (1,0), E2 = (0,1). The x-axis is the line OE1 and the y-axis is the line OE2. Let point P = (a, b), with P1 = (a, 0) and P2 = (0,b). Now suppose that this entire figure is rotated by angle u, with center of rotation = O. We denote the rotated points by adding a prime (e.g., the rotation of E1 is E1', etc.)

Given that the x-axis is rotated to the line OX, with X = (4,3), if P = (3,.2), find the coordinates of all the labeled points in the figure

Suppose that on the plane, O, E1, and E2 and points P, P1, and P2 be as in Example 1, with coordinate axes OE1 and OE2. The entire figure is rotated by angle u, with center of rotation = O, so that the x-axis OE1 is rotated to a line through X = (-2,2). We denote the rotated points by adding a prime (e.g., the rotation of E1 is E1', etc.). If P = (1,-2), find the coordinates of all the labeled points in the figure.

1. Suppose that on the plane, O = the origin = (0,0), E1 = (1,0), E2 = (0,1). The x-axis is the line OE1 and the y-axis is the line OE2. Let point P = (a, b), with P1 = (a, 0) and P2 = (0,b).
2. Now suppose that this entire figure is rotated by angle u, with center of rotation = O. We denote the rotated points by adding a prime (e.g., the rotation of E1 is E1'). Note that distance ratios are the same after rotation.
3. What are the coordinates of E1'?

Hint: Answer uses trig functions of u.

4. What are the coordinates of E2'?

Hint: Answer also used trig functions. What did you learn about rotation by 90 degrees?

5. Since P1' is on line OE1', then P1' is a multiple of E1'. So P1' = kE1' for some number k. [If you prefer position vectors, write OP1' = k OE1'.] What is k in this case when P = (a,b) and what are the coordinates of P1'?

Hint: What is OP1'/OE1'? It is equal to OP1/OE1.

6. Since P2' is on line OE2', then P2' is a multiple of E2'. So P2' = hE2' for some number h. [If you prefer position vectors, OP2' = h OE2'.] What is h in this case when P = (a,b) and what are the coordinates of P2'?
7. The point P' can be obtained by taking a vector sum. The position vector OP' = OP1' + OP2'.

In more geometrical words, the rectangle OP1PP2 is rotated to a rectangle OP1'P'P2'. Since we now know what P1' and P2' are, we can find P' as the missing vertex of the rectangle. Write P as a vector sum.