in-exercises-3-8-find-the-3-times-3-matrices-that-produce-the-described-composite-2-mathrmd-transf-4

Question

In Exercises $3-8,$ find the $3 	imes 3$ matrices that produce the described composite 2 $\mathrm{D}$ transformations, using homogeneous coordinates. 
Rotate points $30^{\circ},$ and then reflect through the $x$ -axis.

Gideon Idumah · Accepted Answer

this question with the root states points 10 degrees and then reflects to the ex exists. So you&#x27;ll recall. That&#x27;s the rotation metrics is given by will sign Phi sine 50 minus sign fied cosign. Fly 0001 So this implies. That&#x27;s it. Let&#x27;s see the grille position abouts. The origin is given by So anyway, you see fire, you put 30. So that implies that school sign of 30 is Rudes three over to sign off. 30 is off on a zero minus sign off. 30 is minus off cause I know Fitzy still with three orbit zoo. I have zero. You have 001 now. So this is the rotation on Danny and also to reflect. True, the ex exists. The metrics for reflection. True, the ex exists will be one 000 minus 10001 or yard a lasagna. Adding your more to plan the second column of the identity by minus woman. Because you are rotating your reflects into the ex exists. So you multiply this on the left. So you multiply these by this because we are doing rotation first, which is days and then we are reflecting. So that&#x27;s why you&#x27;re writing this way. Zero minus off. Woods three over. TSU zero 001 When you do this, you have the answer. Keys earned. Three over. Zoo minus off. Zero minus off, minus words three over TSU 000 Want. So this is the three by three matrix that you get.

Gideon Idumah · Answer

for this question, we need to translate by miners to come on three. So the matrix for trans translation would just be record of Iwan 00010 the translation because at the core column, so they&#x27;ll be minus two theory on then one and then we had told dense killed X coordinates by points aids. So you guys killed exporting to remember that when you&#x27;re done, it&#x27;s yeah, more supplying from the left so the X coordinate is minus. So you&#x27;re multiplying the identity by miners, eh? Just more supplying the X coordinate up. 00 010001 And they are told to skip also the y coordinates by, um, one point suicide. I want 00 01 points, too. 0001 I know what you could have just gets in this too. This treason equivalent to just writing it this way as ah points 800 01 points to 0001 on their multiplied by 10 minus 2013001 which is in quotes so zero points. AIDS 1.20 minus 1.63 point six and don&#x27;t want So this is the three by three matrix that you get

Gideon Idumah · Answer

this was Sugata rooted true for five degrees, about the points to recall my seven. So the rotation metrics we&#x27;re called is given by co sign or five. Sign off I zero minus sign Phi call. Sign off I 0001 So this implies that a 45 degree rotation about the origin is given by you. Do co sign a 45 degrees. You would gets our Earth&#x27;s to overdo Sign off 45 years. Also stupid to zero. You have my notes with two over TSU route to over 2000 Juan. Now we&#x27;re not told to or is it about the 0.3. Calmer? Several. So using the idea from the process problem in this section The process problem in this section that is that the last time the last thing in this in this section we see that&#x27;s two roots. It&#x27;s abouts points to re commerce. Several Remember this is rotating were no reflecting. So what we do is we forced trance leads. Oh, bye. My new story comma minus seven comer then rotates about the origin on dhe translates Buck Bye. Three comma seven. Now what this means in mathematical term is. So this implies. That&#x27;s irritation. A 45 degree of addition. Emma. Right in the way, If 45 degree proposition abouts three comma seven is given by because we are firsts. Um, so the first thing about doing these duds we translated by ministry Come on Several. So we have 100 010 corner. Translation is always a Dick Todd column minus three minus 71 Then we never would sit by foot five degrees, which is urged sue over soon. World&#x27;s Sue over tsu zero minus swords to over. Soon would sue would sue 000 Juan Onda. We not Sean slips back by three. Call my seven snobby 100010 and then 371 So where you want to apply this? You cool off words sue over TSU towards shoe or her too zero minus roots. So over too, boards sue over. Soon, zero three plus two would sue several minus five words to on then one. So this is the matrix to get when you would see its 45 degrees about three. Come on seven

Emily Min · Answer

so using the same triangle, ABC, we want to represent a reflection of these points, a cost X axis, and we want to represent it using matrix form. So first, let&#x27;s represent these points in major ex form. We have point a, uh, negative three comma, one point B at one comma, too. And point c at negative one. Come before and we wanna reflect this across the X axis. So we have to use the property from reflecting the matrix. And the property we use is 10 zero negative one. So what we have to do is we have to multiply these two matrices. Um, but in order to multiply them, you have to use this matrix is the first matrix. And the reason for this is because we&#x27;re gonna be able to playing, um, thes rose by these columns. So you have to have the same number of numbers in the row as we do in each column, and that just wouldn&#x27;t be possible if we had this major expressed. So let&#x27;s carry out this multiplication. We&#x27;re gonna be taking each number in the first row and multiplying it by each number in each column. So you have one times negative. Three his negative three. And we&#x27;re gonna be adding zero times one, which is zero. We have one times one, which is one plus zero times too, which is zero one times negative, one plus zero times for which is zero. Next throw, we&#x27;re gonna be doing the same thing. So we have zero times negative three, which is zero plus negative. One times one, which is negative one zero times one, which is your plus negative one times two, which is negative too. And then zero time thinking of one, which is zero plus negative one times four, which is negative for So let&#x27;s just simplify this. We have negative 31 negative one on the negative one negative. Too negative for So this represents our new set of points after reflecting and crossed the X axis. Um, if you want to write them down as co ordinate points would be negative. Three common negative one one common negative too. And negative. One common negative floor. So that&#x27;s how you complete this reflection using matrix multiplication. And this is the new set of points represented as a matrix

Ankit Gupta · Answer

According to the question, we have given image there. Coordinators are they are given. Gordon airs A is minus three coma. One B is one comer to and C is minus one comma. For when? First we re right in this. Coordinate in a matter form. There is minus 3112 on minus 14 This is a Matics form, according to discussion, The question says that a rotation in 18 to dignity. We know that van a rotation in vanity degree, we multiply with minus 100 minus what? And we find out the image. According to this question, we have these metrics. We find no damage. Onda, we multiply. There is minus three word minus world 1 to 4. Then we multiply. We got the result. Ease three minus lawn minus one minus two one minus four. This is the image metrics we write their coordinates off. Made my taxes. Eight days. Three co minus one B You dad minus one Go. My minus two. I don&#x27;t see you. Dad is one Goma minus four.

Emily Min · Answer

So we have this triangle, ABC and we want to represent a reflection and y equals X on X y plane and want to represent the final co ordinates in the form of a matrix. So, firstly, what meaning to know is that for a reflection across y equals X, we need to use this property. We&#x27;ve been amazing. The matrix 0110 And that&#x27;s a property that never changes When you&#x27;re trying to reflect a set of points across y equals X. So we&#x27;re gonna take this property and we&#x27;re gonna multiply it by the original matrix given to us. So this is gonna be, um, First, let&#x27;s write down the points that we have, actually, So we have a at negative three humble one be at one common too, and see at negative one come before. So to write these points in the form of a matrix, gonna have the exc ordinance on the top room and the white co ordinates on the bottom. So I&#x27;m gonna have negative 31 for 310.8 12 for point beat negative 144 point c and then we&#x27;re just gonna multiply. These two meet receives so where were multiplying Major sees we&#x27;re gonna take the top row of the 1st 1 and multiply it by every number in each column for the second Matrix. So let&#x27;s start with this and we&#x27;re gonna multiply each matching numbers of the first number of the first World by the first number of the first column. So we have zero times negative. Three That zero plus the second number of times the second number. So that&#x27;s one times one plus one. We&#x27;re gonna do the same thing for this one. So it&#x27;s zero times one that zero plus one times two. That&#x27;s two. Then we&#x27;re gonna have zero times negative. One that zero plus one times four. And that&#x27;s for now. We&#x27;re gonna do the same thing for the second row to get our second set of numbers. So we have one times negative. Three. That&#x27;s negative. Three plus zero times. One. That serum one times one that&#x27;s one plus zero times two Is Europe one times negative one this negative one plus zero times for which is zero. So let&#x27;s just simplify these. So we have 124 and negative three one negative. So this is going to represent our new set of points after reflecting on cost Y equals X, so you could leave it as a prime be prime seat front. And if you wanted to write it, coordinate form. We have one common negative three too common one and for common negative one. And this is a prime department. See, Prime. So that&#x27;s how you complete this reflection and how you represent it in matrix form.

Problem

In Exercises $3-8,$ find the $3 \times 3$ matrice…

Problem 6 Easy Difficulty

In Exercises $3-8,$ find the $3 \times 3$ matrices that produce the described composite 2 $\mathrm{D}$ transformations, using homogeneous coordinates.
Rotate points $30^{\circ},$ and then reflect through the $x$ -axis.

Answer

$A=\left[\begin{array}{ccc}{\sqrt{3} / 2} & {-1 / 2} & {0} \\ {-1 / 2} & {-\sqrt{3} / 2} & {0} \\ {0} & {0} & {1}\end{array}\right]$

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David C. Lay, Steven R. Lay, Judi J. McDonald

Linear Algebra and Its Applications

Alayna H.

Kayleah T.

Caleb E.

Michael J.

Absolute Value - Example 1

Absolute Value - Example 2

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$A=\left[\begin{array}{ccc}{\sqrt{3} / 2} & {-1 / 2} & {0} \\ {-1 / 2} & {-\sqrt{3} / 2} & {0} \\ {0} & {0} & {1}\end{array}\right]$

Linear Algebra and Its Applications

Alayna H.

Kayleah T.

Caleb E.

Michael J.

Absolute Value - Example 1

Absolute Value - Example 2

David C. Lay, Steven R. Lay, Judi J. McDonaldLinear Algebra and Its Applications

Alayna H.

Kayleah T.

Caleb E.

Michael J.

David C. Lay, Steven R. Lay, Judi J. McDonald

Linear Algebra and Its Applications