💬 👋 We’re always here. Join our Discord to connect with other students 24/7, any time, night or day.Join Here!

Stuck on your homework problem? This step-by-step video should help.

Try Numerade Free for 30 Days

Like

Report

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). $$\left[\begin{array}{ccc}{1} & {-\tan \varphi / 2} & {0} \\ {0} & {1} & {0} \\ {0} & {0} & {1}\end{array}\right]\left[\begin{array}{ccc}{1} & {0} & {0} \\ {\sin \varphi} & {1} & {0} \\ {0} & {0} & {1}\end{array}\right]$$$$\left[\begin{array}{ccc}{1} & {-\tan \varphi / 2} & {0} \\ {0} & {1} & {0} \\ {0} & {0} & {1}\end{array}\right]$$

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

Algebra

Chapter 2

Matrix Algebra

Section 7

Applications to Computer Graphics

Introduction to Matrices

Missouri State University

McMaster University

University of Michigan - Ann Arbor

Lectures

01:32

In mathematics, the absolu…

01:11

02:52

Write the appropriate rota…

07:35

Rotating Besides translati…

02:01

02:30

Determine the appropriate …

02:49

03:24

PROVE: Matrix Form of Rota…

03:27

02:44

01:33

Compute the determinant of…

01:25

that'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'll use the following identity. We use the following identity to simply fire identity. So simply far now we have That's a ton. Five over. The zoo can be written us one minor school sign of five over. Sign off. I it'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're used to this two identities. So now that implies that's Ivana Cockle. It'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'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't name this one Lemon name this star So we can re write star Oz call sign of five Because we've simplified this to be consigned If I then I have to sign off five zero minus Thanh five over Sue 10001 They're more replied by 100 Minus time off five over Zoo 1000 one Now when you more to apply this to, You'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're gonna have minus sign of five off Tom off. Five. Over soon. Lost one. They're gonna have zero. You're gonna have 001 now when it has simplified this. So that implies that's using identity. Soon you're gonna use the second identity we can rides. Um, okay. Lets you go right to the right things. That'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't. Which I can write us. Pull out in my nose. That'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'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's one minus sign by more supply by sand Five or the two, which is nothing but this. So we've simplified. That'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've been multiplying that the emergencies and we got that It is the order these on What is this mission'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

View More Answers From This Book

Find Another Textbook

Numerade Educator

In mathematics, the absolute value or modulus |x| of a real number x is its …

Write the appropriate rotation formulas so that in a rotated system the equa…

Rotating Besides translating a point, it is also important in computer graph…

Determine the appropriate rotation formulas to use so that the new equation …

PROVE: Matrix Form of Rotation of Axes Formulas Let Z $Z^{\prime},$ and $R$ …

Compute the determinant of each matrix using the column rotation method.…

16:27

For $A$ as in Exercise 11 , find a nonzero vector in Nul $A$ and a nonzero v…

02:38

If $A, B,$ and $C$ are $n \times n$ invertible matrices, does the equation $…

06:00

Let $A=L U$ be an LU factorization. Explain why $A$ can be row reduced to $U…

04:16

Let $A=\left[\begin{array}{llll}{1} & {1} & {1} & {0} \\ {0} &am…

06:35

Exercises $22-26$ provide a glimpse of some widely used matrix factorization…

06:49

Exercises $42-44$ show how to use the condition number of a matrix $A$ to es…

07:39

Balance the chemical equations in Exercises $5-10$ using the vector equation…

00:58

Unless otherwise specified, assume that all matrices in these exercises are …

02:56

Describe the possible echelon forms of the matrix. Use the notation of Examp…

01:41

Compute the determinants in Exercises $1-8$ using a cofactor expansion acros…

92% of Numerade students report better grades.

Try Numerade Free for 30 Days. You can cancel at any time.

Annual

0.00/mo 0.00/mo

Billed annually at 0.00/yr after free trial

Monthly

0.00/mo

Billed monthly at 0.00/mo after free trial

Earn better grades with our study tools:

Textbooks

Video lessons matched directly to the problems in your textbooks.

Ask a Question

Can't find a question? Ask our 30,000+ educators for help.

Courses

Watch full-length courses, covering key principles and concepts.

AI Tutor

Receive weekly guidance from the world’s first A.I. Tutor, Ace.

30 day free trial, then pay 0.00/month

30 day free trial, then pay 0.00/year

You can cancel anytime

OR PAY WITH

Your subscription has started!

The number 2 is also the smallest & first prime number (since every other even number is divisible by two).

If you write pi (to the first two decimal places of 3.14) backwards, in big, block letters it actually reads "PIE".

Receive weekly guidance from the world's first A.I. Tutor, Ace.

Mount Everest weighs an estimated 357 trillion pounds

Snapshot a problem with the Numerade app, and we'll give you the video solution.

A cheetah can run up to 76 miles per hour, and can go from 0 to 60 miles per hour in less than three seconds.

Back in a jiffy? You'd better be fast! A "jiffy" is an actual length of time, equal to about 1/100th of a second.