# How can I describe geometrically the transformation ACA-1 ? (1 Viewer)

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#### Natasha1

If A represents the matrix of a rotation of 45 degrees anti-clockwise around (0,0), B a shear with x-axis invariant and shear factor of 2, and C a reflection in the x-axis.

How can I describe geometrically the transformation ACA-1 ?

My answer is a rotation of 90 degrees anti-clockwise around (0,0) but surely there is more than that? #### HallsofIvy

Natasha1 said:
If A represents the matrix of a rotation of 45 degrees anti-clockwise around (0,0), B a shear with x-axis invariant and shear factor of 2, and C a reflection in the x-axis.

How can I describe geometrically the transformation ACA-1 ?

My answer is a rotation of 90 degrees anti-clockwise around (0,0) but surely there is more than that? There is no B in this problem? Think about what happens to the unit vectors in the x and y directions. Since A corresponds to a 45 degree rotation anti-clockwise, A-1 corresponds to its opposite, a 45 degree rotation clockwise. The vector i is changed by A-1 to the vector $\frac{sqrt{2}}{2}(i- j)$, then by C to $\frac{sqrt{2}}{2}(i+ j)[itex], then by A to j. Okay, that's been rotated 90 anti-clockwise. The vector j is changed by A-1 to [itex]\frac{\sqrt{2}}{2}(i+ j)[\itex], then by C to [itex]\frac{\sqrt{2}}{2}(i- j)$ and finally by A to i. No, that's not a rotation by 90 degrees anti-clockwise! It is a reflection about what line?

#### Natasha1

HallsofIvy said:
There is no B in this problem? Think about what happens to the unit vectors in the x and y directions. Since A corresponds to a 45 degree rotation anti-clockwise, A-1 corresponds to its opposite, a 45 degree rotation clockwise. The vector i is changed by A-1 to the vector $\frac{sqrt{2}}{2}(i- j)$, then by C to $\frac{sqrt{2}}{2}(i+ j)[itex], then by A to j. Okay, that's been rotated 90 anti-clockwise. The vector j is changed by A-1 to [itex]\frac{\sqrt{2}}{2}(i+ j)[\itex], then by C to [itex]\frac{\sqrt{2}}{2}(i- j)$ and finally by A to i. No, that's not a rotation by 90 degrees anti-clockwise! It is a reflection about what line?
line y = x Way to go!

#### Natasha1

That done I need to show that if P' is the image of P under D=BCB-1. If P is not on the x-axis, then PP' is bisected by the x-axis and is at a constant angle to the x-axis, for any choice of P?

I can visually see what's happening and can see that the angle is 90 degrees but how can I show it? :uhh:

#### Natasha1

Could anyone help me with the last question? Please ### The Physics Forums Way

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