Help with this algebra 2 question?

Question

If triangle ABC is defined by the matrix P= -7 4 2

                                                               3 -1 6

 

what are the coordinates of ABC after it has been rotated 90 degrees counterclockwise?

Answer 1

To rotate a matrix 90 degrees do the dot product by the matrix [0 -1; 1 0]

P=[-7 4 2; 3 -1 6]

P'=[-7 4 2; 3 -1 6].[0 -1; 1 0]

P'=[(-7.0+3.-1) ( 4.0+-1.-1) (2.0+6.-1); (-7.1+3.0) ( 4.1+-1.0) (2.1+6.0)]

P'=[(0-3) ( 0+1) (0-6); (-7+0) ( 4-0) (2+0)]

P'=[-3 1 -6; -7 4 2]

Answer 2

Since we don't have information about the rotation reference point, we can assume that it is about the origin.

A 90 degrees ccw rotation about the origin is the same as changing the x coordinate to y and y to -x.

 

Think of it this way.

Visualize the 3hr mark as the x-axis and 12hr mark as the y-axis, the tip of the hour hand as the (x, y) coordinates.

Observe what happens to the (x, y) coordinates when you rotate ccw by 90 degrees.
 

 

For the given matrix P = |-7 4 2| it seems that the format is |x1 x2 x3|  

                                    |3 -1 6|                                        |y1 y2 y3|

 

To use matrix multiplication to perform the rotation, it would be easier to rewrite the matrix as
|x1 y1|

|x2 y2|

|x3 y3|

 

and multiply that by the rotation matrix.

|0 -1|

|1  0|

 

|x1 y1|     |0 -1|               |y1  -x1|
|x2 y2|  * |1  0|      =       |y2   -x2|
|x3 y3|                            |y3   -x3|

 

Your operation would be

 

|-7   3|      |0 -1| 
|4   -1|  *  |1  0| 
|2    6|

 

You obviously need to know how matrix multiplication works in order to get the final answer and understand why the rotation matrix works.