Mapping line y=x finding the angle?

[wduk]

Hello

Quick question that i am confused by.

If i have a rotation g that maps line y = x why is this using a -pi/4 for theta specifically? My brain tells me that surely it should be -pi/2 since if you were at say 0,5 you would then be 5,0 as that is a 90 degree turn counter clockwise so -pi/2 ?

I can't seem to visualise in my mind why it is -pi/4

 

[HallsofIvy]

Hello

Quick question that i am confused by.

If i have a rotation g that maps line y = x why is this using a -pi/4 for theta specifically?

I'm even more confused! What does a rotation mapping y= x mean! Do you mean that g maps line y= x into some other line? If so to what line? I thought for a moment that you meant that y= x was left invariant but that would be a reflection, not a rotation.

My brain tells me that surely it should be -pi/2 since if you were at say 0,5 you would then be 5,0 as that is a 90 degree turn counter clockwise so -pi/2 ?

Are you saying that g(0, 5)= (5, 0)? Again, that sounds like a reflection in the line y= x, not a rotation. What, exactly, does y= x have to do with the rotation?

I can't seem to visualise in my mind why it is -pi/4

 

[wduk]

Yes i was referring to g(0,5) -> (5,0).

It was worded as "Find the matrix of g that maps the line y=x to the x-axis." It then says the rotation is -pi/4 about the origin... but i don't understand how rotating something by -pi/4 would turn (0,5) to (5,0)

 

[stapel]

It was worded as "Find the matrix of g that maps the line y=x to the x-axis." It then says the rotation is -pi/4 about the origin...

Yes; a one-eighth turn clockwise would take the line y = x back down to the x-axis.

but i don't understand how rotating something by -pi/4 would turn (0,5) to (5,0)

It wouldn't. To take points (0, y) from the y-axis to points (y, 0) on the x-axis would require a one-quarter turn.

On what basis had you concluded that the point (0, 5) was of the form (x, x), so that it would lie on the line y = x?

 

[HallsofIvy]

Yes i was referring to g(0,5) -> (5,0).

It was worded as "Find the matrix of g that maps the line y=x to the x-axis." It then says the rotation is -pi/4 about the origin... but i don't understand how rotating something by -pi/4 would turn (0,5) to (5,0)

Good- but that is not what you said before! A rotation, about the origin, through angle \(\displaystyle \theta\) is given by \(\displaystyle \begin{bmatrix}cos(\theta) & sin(\theta) \\ -sin(\theta) & cos(\theta)\end{bmatrix}\).

 

[Electrical]

How to plot y = x

The plot y = x means that for any value of x the y has same value.

For example:
If x = 0, then y = 0;
If x = 1, then y = 1;
If x = 2, then y = 2;
If x = 3, then y =3
and so on.

Now to understand why angle 45 degrees is used here let's consider the plot:
Let's consider a line plotted for x = y

(I tried to rotate the image but there was a problem if you are facing problem reading it please download and then check it.)
Now from the above image, we can draw a triangle and achieve the relationship for Tan theta:

The above relationship is always true for any value of x=y.

So that is the reason that angle is always 45 deg for this line.