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04 composite (ransformation from R? to R? begins by projecting the input onto the line followed by counterclockwise rotation through 459 and ends by reflection abou...

Question

04 composite (ransformation from R? to R? begins by projecting the input onto the line followed by counterclockwise rotation through 459 and ends by reflection about the line y to generate the output Find the standard matrix ofthe composite transformation Is the composite transformation invertible? |ustify your answer: Marks Find u ifv = AS,B.2

04 composite (ransformation from R? to R? begins by projecting the input onto the line followed by counterclockwise rotation through 459 and ends by reflection about the line y to generate the output Find the standard matrix ofthe composite transformation Is the composite transformation invertible? |ustify your answer: Marks Find u ifv = AS,B.2



Answers

Find the standard matrix for the reflection of $R^{2}$ about the stated line, and then use that matrix to find the reflection of the given point about that line. The reflection of (1,2) about the line that makes an angle of $\pi / 4\left(=45^{\circ}\right)$ with the positive $x$ -axis.

All right. The key idea in this problem is if you are going to do one transformation, say T of A. And follow it with another transformation, say T F B, then that has a standard matrix of B times A. Okay, so the key here is we're going to do some matrix multiplication. And the tricky part is we have to get the order correct. Okay, So in this first problem, What we're gonna do first it says is rotate 90°. So that is my t of a. And then we're going to reflect in the line Y equals X. So this is my T of B T C B. Okay. And now we know If I'm going to rotate 90° that has a matrix, it looks like this Co sign 90° Negative sign of 90° Sign of 90°. And then the co sign of 90°. Yeah. And if they simplify that matrix, I get the matrix 0 -110. Okay, so that's my a matrix for tfb reflecting in the line Y equals X. We know that has the matrix 0. 10. Okay. And so now To get the matrix for this whole transformation during the 1/1 and then the second. What we need to do is we need to take B times A. In that order. So I need to take the 0110. And multiply that. Bye. The Rotation Matrix which is 0 -110. Do that matrix multiplication. We're going to get one zero zero negative one. Okay. And so that is the matrix for the transformation where you first rotate 90° and then reflect in the line y equals x. Okay. Okay. Super. Let's do another one. We're going to project onto the Y axis. So that's what we're gonna do first and then after that we're gonna do a contraction with a factor of K equals a half. So that is my B. Matrix. Okay? So if we remember projecting onto the Y axis has a matrix Of 0001. And contraction matrices. It's just a scalar multiple times the identity. So a half 00 a half. Because my factor is a half. Okay. And again I need to just make sure the order is correct. I want to do a first. So I'm going to put that on the right and then multiply by B. Okay, So I need to find out what B times A. Is. So that is a 0.50 a half. There's my baby Times A 0001. Let's do that matrix multiplication. We're going to get zero zero. This row times, this column is going to get me zero and then this real times this column is going to get me a house. And so there is the standard matrix for the composite transformation project onto the Y, followed by a contraction. With a factor cake was a half. Okay. Alrighty. Let's do one more here. We're going to reflect about the X axis. So that's the first thing I want to do. It's going to be my A Followed by a dilation with cake was three. That's my baby. And now there's a third one here followed by a rotation of 60°. It's okay. Nothing too exciting. That's just going to be a third matrix. And then what I need to do is I need to do A and then multiply that by beyond the left and then multiply that by see on the left. So I need to calculate C times B times A. Okay, so let's see if we can just write this down Matrix C. is a rotation of 60°. So that looks like co sign of 60°. The opposite of the sign of 60°. The sign of 60° And then the co sign of 60°. So that is my C. Matrix. That's the thing I'm going to do. Last dilation with K equals three is my B. Matrix. So that is just going to be three times the identity matrix. So 3003 There's my baby. And then finally, the first transformation I'm gonna make is I want to reflect about the X. Axis. And so that has a matrix. It looks like 100 -1. Yeah. And so again, the order is backwards, right? Because again, if I was going to be multiplying by a vector here, I want to multiply by the A. First and then the B. And then the sea. Okay. All right, so let's do this multiplying of three matrices. And remember you can only multiply two things at a time. So I'm going to multiply these together first. Okay. And if I multiply those together, I get the matrix three. 00, -3. Okay, so that's my matrix B times A. And then let me simplify. See here while we're going through here, Co sign of 60° is a half negative sign of sixties negative Route 3/2 sign of 60 is route 3/2, And co sign of 60 is a half. So that's my C matrices simplified. Okay, now let's multiply these two together. Okay, So I'm going to get 3/2ves and then I'm going to do this road times this column. And so I will get uh negative negative squared of 3/2, which is going to be three squared of 3/2. Okay, from here, Times here, now we're going to take this road times this column. So I'm going to get three square roots of 3/2, and then finally the bottom row times the last column, Which is going to get me a -3 house. Right? Sounds good order. Really important when we're working with matrices, make sure you get the order correct. Thanks.

Hello there. In this exercise we have this .34 and we have a line find that passed through the origin that forms an angle of 60° with respect to the X. Axis. So this is morally is kind of the representation of these line. And In this extra says we need to work with a reflection of this .34 about this line. So basically we need to obtain a point that is over here. That is the geometric idea. So to do that we need to use the reflection matrix. So let's remember that the reflection matrix even by some angle Theta is equals to go. Sign up to feta sign of two theater sign of two seater and minus co sign of to feed. So what we need to do is just replace the values and remember that An angle of 60° is also equivalent to pi three Radiance. So in that case we need to work with the matrix given by power thirds. And these results into the matrix given by 1/2 times. Mhm. The matrix minus one squared three squared of three and one. Okay, so this is the matrix that we need to multiply to the to the to this vector to obtain the reflection with respect to that line. So the only thing that we need to do now is just multiply these matrix with that point. So here just copying again. The matrix Times Vector 3 4. The result is one half times the vector for Times Square root of three mines 3, three times square the three plus four.


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