a-rotation-in-mathbbr2-usually-requires-four-multiplications-compute-the-product-below-and-show-that

Question

A rotation in $\mathbb{R}^{2}$ usually requires four multiplications. Compute the product below, and show that the matrix for a rotation can be factored into three shear transformations (each of which requires only one multiplication). 
$$
\left[\begin{array}{ccc}{1} & {-	an \varphi / 2} & {0} \ {0} & {1} & {0} \ {0} & {0} & {1}\end{array}ight]\left[\begin{array}{ccc}{1} & {0} & {0} \ {\sin \varphi} & {1} & {0} \ {0} & {0} & {1}\end{array}ight]
$$
$$
\left[\begin{array}{ccc}{1} & {-	an \varphi / 2} & {0} \ {0} & {1} & {0} \ {0} & {0} & {1}\end{array}ight]
$$

Gideon Idumah · Accepted Answer

that&#x27;s compute the products below and we starts I by writing out the Mitchell sees off 100 my not minus time five or would sue 10 00 Juan more supplied by one sign. Phi zero 010 001 on Ben one man mine istan Fight over the zoo and I have 00 on This is 10001 When you multiply this too you have This is called Sue. Want my nose, son? Five? Or did sue more supplied by son? Five Sign fi zero. Then you have mine. Ists fight Will that Soo wan 000 Juan. And then you bring this down. You have 100 my nurse Ton five over. Tsu want 0001 Now we want to write these. We&#x27;ll use the following identity. We use the following identity to simply fire identity. So simply far now we have That&#x27;s a ton. Five over. The zoo can be written us one minor school sign of five over. Sign off. I it&#x27;s can also Bharat Singh Osl Emma using red Pan. Oh, good pen. This can also be written us sign five over one plus co signed five. So we use. We&#x27;re used to this two identities. So now that implies that&#x27;s Ivana Cockle. It&#x27;s D. So let me name this identity. Juan lets me name this identity Sue. So that implies that using I dented see one we have that we can write sworn miner&#x27;s son five over Zu Sein off I as want miners Then I have a one minor school sign off five over. Sign off I more supplied by sign off five So discusses days you have one minus one. Plus this this is just co sign off five So this implies that I can re writes lets me live this equation One not implies that recover writes one us we can re writes You can let me use a different number since I didn&#x27;t name this one Lemon name this star So we can re write star Oz call sign of five Because we&#x27;ve simplified this to be consigned If I then I have to sign off five zero minus Thanh five over Sue 10001 They&#x27;re more replied by 100 Minus time off five over Zoo 1000 one Now when you more to apply this to, You&#x27;re gonna have call Sign off. I sign of fi zero co signed off. I more supplied by minus son off five over. Soup minus time off five or the zoo and they&#x27;re gonna have minus sign of five off Tom off. Five. Over soon. Lost one. They&#x27;re gonna have zero. You&#x27;re gonna have 001 now when it has simplified this. So that implies that&#x27;s using identity. Soon you&#x27;re gonna use the second identity we can rides. Um, okay. Lets you go right to the right things. That&#x27;s cool. Sign of five off. Minus time off. Fight over. Zoom minus or fight Or the zoom. First of all, I confortable pull out in town five and Sue, if I pulled that out, what I have left will be minus cause sign off five from here. And then miners won&#x27;t. Which I can write us. Pull out in my nose. That&#x27;s minus town five or the zoom off one loss. Co sign off five. Now, if I use the identity soon. Dobby, I can write, son, fight over tsu as signed five of our one possible sign of five. So does just minus Sign five over one plus co Sign off. Five more supplied by one plus co sign before. So these were constant duties on then. This is just minus sign five gold. So we&#x27;re simplified these now, what is this simplification for these. If you go back, this is This is this is gonna write us one. I can already there&#x27;s one minus sign by more supply by sand Five or the two, which is nothing but this. So we&#x27;ve simplified. That&#x27;s to be a call to co sign off. Five. Saddam is I can rewrite Let me name distribution Star Stop So we cover rights He writes Star Star Oz the Doctor The old soon call sign of five Sign of 50 My no sign of fire course Sign off 500 There one. So what is 1/4? These We&#x27;ve been multiplying that the emergencies and we got that It is the order these on What is this mission&#x27;s? This is the transformation corner. This is just the rotation matrix. This is the transformation. Many trees mean more genius quarterly needs for your audition Moochers for a fish sandwich. So we have answered the question

Ankit Gupta · Answer

we have given an increase in dirties excess square plus for X Y minus two y squared minus one equal to zero. We have to find the appropriate rotates and formula so that in a rotating system that Chris and has no express my restaurant, there&#x27;s why we compare it with this genetic face and we get A is equal to one. B is equal to four. C is equal to minus two. Now court rotator is equal toe a minus. C divided by be That gives us one minus minus two divided by, for that is equal to three, divided by four. Since Trita is always a cured and court through, Tita is positive that survive rotator is in first Courtland. It means the third side off right triangle is found using the vital wouldn&#x27;t from there is three squared plus four squared is equal to artist grab or our squad is equal to 25. We get our is equal to five. That&#x27;s why course rotator will be equal toe three divided by fight Now using question treat of you find the very top scientist and question because Santa is equal to one miners coach toe traitor divided by that is equal to the square root off one minus study, divided by five. Ready to bite toe up recently because and we get sensitize equals the square root of February by five, and question is equal to the square. Root off one plus course. Rotator divided by that is equal to the square. Root off one plus three, divided by five, divided by that is equal to the square root of five. Divided by five. We know that X is Equal Toe Express course, cheetah minus wide as saying Tita replacing the world upside return portion We get X deaths multiplied by two Scotto fired word by five minus y days multiplied by the square root of five, divided by five after simplifications and we get X is equal toe Esquire Roto fire multiplied by two exodus minus Why death Divided by fire and by is equal toe xs. Scientific data. Plus Why does course Teuta that is equal to X desk when deployed by a square root of five Divided by five. Plus why does multiplied by two square Oto five divided by five are presently because and we get why is it worth to square root off fire X deaths plus two y death directed by five. It means ditch toe are the appropriate rotation formula so that in a rotated system, the eggs isn&#x27;t has no express my desk.

Christine Anacker · Answer

hurts. So we&#x27;re doing the rotation matrix instead of a translation. So they&#x27;ve given us the 0.64 and we need to do this. So we need to go matrix r times X. So have 3/2 zero native or three or two have zero 001 times are point 64 one. So I&#x27;m going to cheat the system sickness in my calculator, especially because I&#x27;m dealing with roots and what not going into my matrix. I&#x27;m doing a three by three. Have a tape this in so 1/2 negative root. 3/2 zero. I don&#x27;t. Three over to 1/2 0 001 Make sure I have these entered. Right? I heard three or two aereo Now they&#x27;re entered. Right. Alright, so I&#x27;ve got that done. Now I need to go into my matrix again. And eight Edit My Oops. Not that one. I need to edit my second re tricks to be a three by one Matrix that 64 and one. So then I could go matrix a times matrix B on. I&#x27;ll get my answers now because I end up dealing with does meals here. I&#x27;ll quickly just write out what we did, so we would have gone 1/2 times six plus negative or 3/2 times for plus zero times one, uh, 3/2 times six less, 1/2 times for plus zero times. One that zero times six plus zero times. For unless Ciro one time is one. So if you do this on a calculator, you will get decibels. If you do this typed out, we end up with half of six, which is three on negative 2/3 as their one answer and three Route three plus two is our 2nd 1 and one as her 3rd 1 So what that tells us here is that a 0.64 got translated when it moved 60 degrees three, but is to root three and three, Route three plus two. It&#x27;s that is part a Part B wants us to find our rotational inverse. So if you already still have it typed into your calculator, you can super quickly just take that and, um, go from there. If you don&#x27;t have it typed into your calculator, you want to do it the longhand way. Then we&#x27;re gonna have to take our regular matrix on, put it next to an identity matrix. And so from their awesome did not East scenarios are in. Take that. Put it next to our identity matrix. And so from there, when we try to do this quickly So I&#x27;m gonna take row one and multiply by two. Have it with one. They divert three zero 200 actually gonna multiply this one by two as well. So we&#x27;re 310 0 to 0 001001 Alright. Awesome. And then I went to take root to and subtract. Three times were one, So would they hurt? 30 is not changing to there. There was not changing in. My third row is not changing. All right, so hurt three of minus for 30 would plus three is four zero zero minus two or three is negative. Two or three tu minus zero is too. And I have a zero. No, I&#x27;m going to take my second rule and divide it by 1/4. So one they divert 30 010 Syria 01 Choose. Your zero has become negative. Route 3/2 1/2 0001 All right last step doing it the long way. So I am going to take my row one I have three times wrote to So one plus zero is one negative. Route three plus route 300 plus 00 uh to minus three. House this negative 1/2. Uh, zero plus route three or two is Route 3/2 000 So there is our inverse matrix for our rotation. So are adverse. His native 1/2 3/2 zero. Thank you. Divert three or two. 1/2 0 00 and one.

Ankit Gupta · Answer

we have given any grease in there. It is X squared plus for X y plus Vice squad minus Terry equal to zero. We have to write the appropriate rotates and formulas so that in a rotated system disagrees and has no actually. That&#x27;s why Estrin no compared this crescent with this gender decrease and we get a world one be equal toe for and Sea World. We get court through. Tita is equal toe aim in a CD right away. Be there is equal to one minus one day, whereby for that is equal to zero. It means rotator is equal to 90 degree, or Twitter is equal to 45 degree. It means the angle of protests and is 45 degree. Now we will find the value off X and Y X is equal to X days course off for defense. Big Guerry minus Why does sign for 40 five days really are per simplifications and vigor. Excess multiplied by one upon route toe minus y days multiplied by one upon route taken upon Ruto as a common from and replacement upon to Beirut to divide by two. You get this so replace X squared upto divided by toe accidents minus right this Now we will find them. Look why that would be the place. Why is equal to Exodus sign or four different degree? Plus why discourse off 45 degree are presently because And vigor square root off, divided by toe xs Plus why this It means, if very place why by a square root upto upon toe divided by FDs Plus why days and X by square root up to do it right With this minus by just then then in a rotating system the sequester will not have any accidents Why does trump

Chris Trentman · Answer

were given an equation were asked to determine appropriate rotation formulas, so the new equation has no X Y turn. The equation is X squared minus four X y plus y squared minus three equals zero. By comparing this equation to the equation for a General Connick section, we see that a is the coefficient of X squared, which is one B is the coefficient of X y, which is negative for and C is the coefficient of y squared, which is one. So we have that he is not equal to zero. Therefore, by a serum from the book. Do you have that? The appropriate angle Feta satisfies co tangent of tooth data equals a minus C over B, which is one minus one over negative four or zero over negative four. Just simply zero. This implies that co sign must be zero. So we get that to theater is equal to 90 degrees, or the theater is equal to 45 degrees by another theory, um, in the book formulas for rotation are X equals X Prime Times Co. Sign of 45 degrees, minus white prime of sign of 45 degrees. And why is ex prime sign of 45 degrees plus y prime co sign of 45 degrees. Since signing co sign of 45 degrees is routes over to we have that X is equal to root two over two times Ex prime minus y prime. And why is equal to root two over to times Ex prime Plus why crime? This is our answer.

Foster Wisusik · Answer

Okay. This question wants us to find the rotation formulas that could be used to eliminate the X Y term from this Konik. So we&#x27;ll always start by using the fact that the co tangent of Tooth Ada is equal to K minus C all over b. So a is the X squared coefficient. See, is the Y squared coefficient and be is the X Y coefficient. So we just get zero over negative four or zero. So this tells us that the co tangent of Tooth ada is equal to zero. So co sign tooth ada over signed tooth Ada must then be zero and multiplying by. Sign on both science, we get co sign tooth data is equal to zero. And for our rotations, the only angle that would satisfy that would be when tooth data is equal to hi over to. So that means data is pi over four, which is 45 degrees. So now that we found our angle, we can write our coordinate conversions, so X is equal to ex prime coastline data minus why prime signed data and why is equal to ex prime sign data, plus why prime coastline data and this is actually pretty simple because co sign and sign of pi over four are both just route to over to. So the formulas just produced nicely. Toe acts is equal toe one over route to times ex prime minus y prime. And then why is equal to one over route two times ex prime plus y prime. And you could write this as route to over two also, But just if we&#x27;re going to be squaring it later, I think this is a more convenient for him to look at.

Problem

The usual transformations on homogeneous coordina…

Problem 12 Medium Difficulty

Answer

$\left[\begin{array}{ccc}{\cos \varphi} & {-\sin \varphi} & {0} \\ {\sin \varphi} & {\cos \varphi} & {0} \\ {0} & {0} & {1}\end{array}\right]$

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$\left[\begin{array}{ccc}{\cos \varphi} & {-\sin \varphi} & {0} \\ {\sin \varphi} & {\cos \varphi} & {0} \\ {0} & {0} & {1}\end{array}\right]$

Linear Algebra and Its Applications

Catherine R.

Alayna H.

Maria G.

Kristen K.

Absolute Value - Example 1

Absolute Value - Example 2

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

Catherine R.

Alayna H.

Maria G.

Kristen K.

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

Linear Algebra and Its Applications