Vector Reflection Across y=2x: Solving with Rotation and Change of Bases

[fattycakez]

Homework Statement

Find the (exact) reflection of the vector v = (5, 1) across the line: y = 2x.
Hint: A sketch of v and the line may suggest an approach.

Homework Equations

The Attempt at a Solution

I found the matrix
-3/5 6/5
4/5 2/5
which seems like it gives the reflection across y=2x

But my question is: is there way to do this by rotating the axes and changing bases? (I'm pretty sure this is what the assignment is asking me to do)
I'm having a hard time visualizing it since no angle is given to put into the rotation equations for R²
i.e.
x'=xcosθ +ysinθ
y'=-xsinθ+ycosθ

Any help is greatly appreciated :)

 

[SammyS]

Homework Statement

Find the (exact) reflection of the vector v = (5, 1) across the line: y = 2x.
Hint: A sketch of v and the line may suggest an approach.

Homework Equations

The Attempt at a Solution

I found the matrix
-3/5 6/5
4/5 2/5
which seems like it gives the reflection across y=2x

But my question is: is there way to do this by rotating the axes and changing bases? (I'm pretty sure this is what the assignment is asking me to do)
I'm having a hard time visualizing it since no angle is given to put into the rotation equations for R²
i.e.
x'=xcosθ +ysinθ
y'=-xsinθ+ycosθ

Any help is greatly appreciated :)

What angle does the vector, <5, 1>, make with the line y = 2x ?

 

[fattycakez]

What angle does the vector, <5, 1>, make with the line y = 2x ?

Man I'm slow, it makes an angle of 52.125! When I use that and the (5,1) in the rotation equations it looks like its reflecting in the wrong direction
(4th quadrant rather then second quadrant)
The new vector appears to be at a 90 degree angle with y=2x, do I need another rotation or something like that?
Thanks :)