Conic Rotation

klaus113 · #1 $Old$ November 13th, 2009, 03:53 PM

Hi

Please help me. I want to know how to find the coordinates of a circle after they have been rotated some degrees.

One problem I read wrote this:

"Shown in the standard (x,y) coordinate plane below, pint P (6,6) is rotated 90 degrees clockwise about the orgin (2,3). What are the coordinates of P after the rotation has been complete?"

I know the answer is (5,-1), but I don't know why that is the answer or how to solve the problem for other questions!

Grandad · #2 $Old$ November 13th, 2009, 11:27 PM

Hello klaus113

Welcome to Math Help Forum!

Quote:

Originally Posted by klaus113 $View Post$

Hi

Please help me. I want to know how to find the coordinates of a circle after they have been rotated some degrees.

One problem I read wrote this:

"Shown in the standard (x,y) coordinate plane below, pint P (6,6) is rotated 90 degrees clockwise about the orgin (2,3). What are the coordinates of P after the rotation has been complete?"

I know the answer is (5,-1), but I don't know why that is the answer or how to solve the problem for other questions!

Think about any point whose coordinates are $(x_1,y_1)$ - see the attached diagram. Its horizontal and vertical distances from $(2, 3)$ are $x_1-2$ and $y_1-3$ , respectively. After a rotation through $90^o$ clockwise about this point, these will be the vertical distance below and the horizontal distance to the right of this point.

So the coordinates of the image of $(x_1,y_1)$ after this rotation are

$(2+y_1-3, 3 -[x_1-2])$ ; i.e. $(y_1-1, 5-x_1)$

This gives the general formula you're looking for, and confirms that $(6,6) \mapsto (5, -1)$ .

Grandad

Soroban · #3 $Old$ November 14th, 2009, 06:39 AM

Hello, klaus113!

Quote:

Point $P(6,6)$ is rotated 90° clockwise about the point $Q(2,3)$ .
What are the coordinates of $P$ after the rotation is completed?

A sketch is always a good idea . . .

Code:

        |         P
        |           o (6,6)
        |           :
        |           :3
        | Q o - - - +
        |(2,3)  4
        |
    ----+-----------------
        |

Going from $Q$ to $P$ , we move 4 right and 3 up.

Code:

        |
        |(2,3) 3 
        | Q o - - +
        |         :
        |         :4
    ----+---------:----
        |         o P'
        |

Going from $Q$ to $P'$ , we move 3 right and 4 down.
. . Do you see why?

Now we know the location of $P'$ .