Some Math Problems (Matrices): point, fcn transformations, rotations, reflections

[tyrowe01]

Here are some problems I need help solving, preferably worked through or at least with an explanation

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18) Consider the transformation:

. . . . .\(\displaystyle \begin{cases}x'\, =\, \dfrac{3}{5}\,x\, +\, \dfrac{4}{5}\, y \\ y'\, =\, \dfrac{4}{5}\, x\, -\, \dfrac{3}{5}\, y \end{cases}\)

Perform this transformation to the unit square given by the points O(0, 0), A(-1, 0), B(-1, 1), and C(0, 1). List any invariant points, explain your answer. Is sense preserved, explain your answer.

19) Find the image of the point \(\displaystyle \, \scriptsize{ \left(\dfrac{1}{\sqrt{\strut 2\,}},\, -\sqrt{\strut 2\,}\right) }\,\) under a rotation through an angle of 225 degrees.

20) Find the image of the equation x² + y² = 4 under the rotation of \(\displaystyle \, \dfrac{\pi}{3}\,\) about the origin.

21) Find the image of the equation y = 3x - 2 under the reflection across the line \(\displaystyle \, y\, =\, \dfrac{3}{2}\, x.\)

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you don't have to do all of them or in any order, anyone that is able to answer any would be super helpful (the writing in the corner can be ignored it is from a one I was solving above). Thank you guys!


https://gyazo.com/be2724bd4a22be8e381fdbaa55e61ca7 (here is a screenshot just in case you can't see it)

 

[Deleted member 4993]

View attachment 8100

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[HallsofIvy]

Number 18, at least, is very simple (and does not require using matrices). You are given the linear transformation \(\displaystyle x'=\frac{3}{5}x+ \frac{4}{5}y\), \(\displaystyle y'= \frac{4}{5}x- \frac{3}{5}y\) and asked to find the image of the unit square with vertices (0, 0), (1, 0), (1, 1), (0, 1) under that linear transformation.

Just put the given numbers into that formula: x= 0, y=0 is transformed to \(\displaystyle x'= \frac{3}{5}(0)+ \frac{4}{5}(0)= 0\), \(\displaystyle y'=\frac{4}{5}(0)- \frac{3}{5}(0)= 0\). x= 1, y= 0 is transformed to \(\displaystyle x'= \frac{3}{5}(1)+ \frac{4}{5}(0)= \frac{3}{5}\), \(\displaystyle y'=\frac{4}{5}(1)- \frac{3}{5}(0)= \frac{4}{5}\). x= 1, y= 1 is transformed to \(\displaystyle x'= \frac{3}{5}(1)+ \frac{4}{5}(1)= \frac{7}{5}\), \(\displaystyle y'=\frac{4}{5}(1)- \frac{3}{5}(1)= \frac{1}{5}\). And x= 0, y= 1 is transformed to \(\displaystyle x'= \frac{3}{5}(0)+ \frac{4}{5}(1)= \frac{4}{5}\), \(\displaystyle y'=\frac{4}{5}(0)- \frac{3}{5}(1)= -\frac{3}{5}\).

The last part of 18 asks if "sense" is preserved. Do you understand what "sense" is? If we "move" around the unit square in the direction in which the vertices are given, (0, 0) to (1, 0) to (1, 1) to (0, 1) and back to (0, 0), we move counter-clockwise. If we move around the transformed square in the order of those transformed vertices, (0, 0) to (3/5, 4/5) to (7/5, 1/5) to (4/5, 3/5) and back to (0, 0) are we moving clockwise or counter-clockwise? (It might help to graph these points.)

 

[stapel]

18) Consider the transformation:

. . . . .\(\displaystyle \begin{cases}x'\, =\, \dfrac{3}{5}\,x\, +\, \dfrac{4}{5}\, y \\ y'\, =\, \dfrac{4}{5}\, x\, -\, \dfrac{3}{5}\, y \end{cases}\)

Perform this transformation to the unit square given by the points O(0, 0), A(-1, 0), B(-1, 1), and C(0, 1). List any invariant points, explain your answer. Is sense preserved, explain your answer.

19) Find the image of the point \(\displaystyle \, \scriptsize{ \left(\dfrac{1}{\sqrt{\strut 2\,}},\, -\sqrt{\strut 2\,}\right) }\,\) under a rotation through an angle of 225 degrees.

20) Find the image of the equation x² + y² = 4 under the rotation of \(\displaystyle \, \dfrac{\pi}{3}\,\) about the origin.

21) Find the image of the equation y = 3x - 2 under the reflection across the line \(\displaystyle \, y\, =\, \dfrac{3}{2}\, x.\)

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Here are some problems I need help solving, preferably worked through or at least with an explanation

you don't have to do all of them or in any order...

We "don't have to do" any of them; you do. We can help, but we'll first need to see what you have tried. Or, if you're really needing lesson instruction first, please specify over which topics, so we can provide lesson links.

But this isn't one of those "cheetz" sites that'll do your homework for you. For further info, please re-read the "Read Before Posting" thread. Thank you!