use-the-terminology-from-section-82-a-let-tleftleftbeginarrayr-1-0endarrayrightleftbeginarrayl2-3end

Question

Use the terminology from Section 8.2
a. Let $T=\left\{\left[\begin{array}{r}{-1} \ {0}\end{array}ight],\left[\begin{array}{l}{2} \ {3}\end{array}ight],\left[\begin{array}{l}{4} \ {1}\end{array}ight]ight\},$ and let $\mathbf{p}_{1}=\left[\begin{array}{l}{2} \ {1}\end{array}ight], \mathbf{p}_{2}=\left[\begin{array}{l}{3} \ {2}\end{array}ight], \mathbf{p}_{3}=\left[\begin{array}{l}{2} \ {0}\end{array}ight],$ and $\mathbf{p}_{4}=\left[\begin{array}{l}{0} \ {2}\end{array}ight]$.
Find the barycentric coordinates of $\mathbf{p}_{1}, \mathbf{p}_{2}, \mathbf{p}_{3},$ and $\mathbf{p}_{4}$ with respect to $T$.
b. Use your answers in part (a) to determine whether each of $\mathbf{p}_{1}, \ldots, \mathbf{p}_{4}$ in part (a) is inside, outside, or on the edge of $\mathrm{conv} T,$ a triangular region.

Lucía Guerrero · Accepted Answer

for part A of this exercise. We&#x27;re going to use the homogeneous forms off P I on, um, the eyes. So to find the very central coordinates, we have to write the homogeneous form of B I as a linear combination off the homogeneous forms off 31 52 53 on. We can do that by reducing the following augmented matrix. So the first column is gonna be the homogeneous form corresponding off to be one, which is one minus one cereal on, then 123 on, then one for one on. Then on the right hand side, we&#x27;re going to put the machine is form of B one, which is 1 to 1. Then the machine is from B two. That&#x27;s it through. So once we reduce this matrix we obtained a fold. First row is gonna be one 00 to 60 0. R. One third. She wrote one half. Well, huh? The second row is going to be 010 1/6 one half when this 1/4 on 9 12, which is also 3/4 on the last row is gonna be serious. 011 half one half 3/4 on minus 1/4. No. Each of these columns represent the virus. Entry coordinates off the P ice Now for part B. We have to say if this points are inside or outside the complex haul off T. So let&#x27;s take a look at this column which corresponds to be one all off its entries which could respond to the very century coordinates of P one with respectively 1, 15, 53. All of them are positive. They sum up to one eso. This means that this point B one is inside the triangular region. That is the convicts whole off t. Now this column represents speak to on. We can see that there is a entry that is zero on This means well the rest of the entries of this column are positive. So that means that p two is on the edge of this triangular region that represents the convicts Hall of tea. On finally the last two columns which represent Petri MP four, which correspond to be three mp four, we can see that they have negative coordinates. This is enough to say that they are outside off the triangular region, which is the comics hall of tea

Mrinal Rana · Answer

So after the transformation, points RST but say they become our prime s prime and t prime. And those are gonna have coordinates. Negative one negative. 12 negative. Three negative. Nine on dhe. Negative seven. Negative seven. Those air. The coordinate points after the transformation.

Jose Hannan · Answer

Hello, everybody. In this video, I&#x27;m gonna be showing you how to sell of exercise 36 in Chapter 13 section for calculus early. Transcendental, that this problem provides us with three points. A with coordinates. Negative one negative five. Negative three. Be with coordinates. Negative. Free. Negative to negative one is Willacy with coordinates. Zero negative five negative one. The problem. Masses To find the area of the triangle that is formed by these three points to begin, let&#x27;s first acknowledge that this triangle part of a larger parallelogram that&#x27;s drawn like this, which has twice the area as the triangle interested in that is this part we just headed here. That&#x27;s the same area that as a triangle we&#x27;re investigating. So to analyze this parallelogram, what we can do is find the vectors a B and a C they be is equal to They get a three minus negative one negative two minus negative five as well as negative one, minus negative three. So what we get is negative. Three plus one is negative. Two negative two plus five, just three and negative one plus three to the vector A B next. We also find what a C is they see is equal to zero minus negative. One negative five minus negative five as well as negative one minus negative. Three zero minus negative one. It&#x27;s just one. They get a five minutes Negative. 50 and negative One plus three is too. So we get 102 for the vector A C. So no, notice that with these vectors, Director A B forms this pair of parallel sides of the parallelogram while a C forms this hair of parallel sides with a parallelogram. And so we can say that the victors a B and A C form this parallelogram and then by here on 13.3 we have at the area of this parallelogram which is two times of the area of the triangle that we&#x27;re interested in, is equal to the magnitude of the cross product between the vectors a, B and A C which form the parallelogram. So what? We want to dio it&#x27;s fine. This cross product a be cross a C and then find its magnitude to determine the cross product. I&#x27;ll be using the determinant method as described in fear in 13.6. To do this, we form a three by three Major With the standard unit vectors in the top row, the coordinates of a B in the second row get up to 32 and in the coordinates of a C in the bottom row one zero and to no, the determinant of this matrix is equal to i times the determinant the bottom right sub matrix here three to zero and to it&#x27;s attracted from that we have a similar quantity for J which is J times the determinant of this sub matrix formed by this bottom left sub column. And this bottom right, some column negative too to one in two. And added to that we have a likewise quality for K which is K times the determinant of this bottom left sub metrics here get up to three, one and zero. So our next step is to compute the determinants of each of these three sub matrices. The first term we have three times to which is six minus zero times two, which is zero. So we get six. I subjective from that we get negative two times two negative four minus one times two, which is to times J and then added to that we have negative, too. I&#x27;m zero, which is zero minus three. So we get minus three care. So once we simplify that middle Herman, the parentheses What we get is negative for minus two, which is negative. Six. As we&#x27;re subtracting this, we&#x27;re actually adding six j and then once again subtracting we care. And so what we get is that this is a vector with components six six and negative three for a B cross. Easy that to make the ensuing calculations a little bit easier. What we can dio Let&#x27;s pull the factor of three from the components of this vector. We have three as a scaler on the outside times a vector to to negative one. And this is just so that we don&#x27;t have to deal with adding together larger squares when we compute the magnitude of this vector, but that we&#x27;re ready to compute this magnitude. You have that the magnitude of a B cross a C which is equal to the area of the parallelogram which is equal to two times The area of our triangle is equal to the square root. Three squared times a quantity two squared was two squared. What&#x27;s negative. One squared. We can pull the three out of the square root, and we get that this is equal to three times the square root before what&#x27;s for plus one now four plus four plus one is he called the nine we get that this is three through nine. Then the square root of nine is three. We got three times three, which is equal to nine for the area of the parallelogram. And therefore the area of the triangle is gonna be half this area nine divided by two. And so we arrived at our answer for the area of the triangle. And that&#x27;s how you solve problem 36.

Melissa Stefan · Answer

to determine whether these three burgesses will form a triangle. We first need to find the distance between each para ver theses that will translate to the side lengths. And then once we have those distances, we can compare the triangle inequality theorem for the three inequalities to make sure that the sum of any two side just greater than the third side. So starting with finding our distance between the pairs of vergis ease, we&#x27;re gonna use our distance formula that I have put up here as a reminder. And using Vergis is R s to find the length of the segment we have. RS is equal to the square root of negative three minus one squared plus negative 20 minus negative four or negative 20 plus four squared and they were just going to simplify it. And this is going to be the squared of negative four squared plus negative 16 squared, which is equal to the square root of 16 plus 256 which finally simplifies to the square root of 272. I&#x27;m gonna leave everything in radical form so that we don&#x27;t have any rounding errors. Let&#x27;s move on to the segment form Beiber Theses S T. So S T is going to be equal to the square root of five minus negative three or five plus three squared plus 12 minus negative. 2012 plus 20 squared, which is equal to the square root of eight squared plus 32 squared, which is equal to the square root 64 plus 32 squared is 10 24 and that finally simplifies to the square root of 10 88. Just one more to go and we need the segment length of party so R T is equal to the square root of five minus one squared plus 12 minus negative four, which is 12 plus four squared simplifies to the square root of four squared plus 16 squared, which is equal to the skirt of 16 plus 256 which is equal to the square root of 272. It appears we have to side lengths that are equal. We&#x27;re still going to have to see if are triangle. Inequality will hold, so to form a triangle, remember, we have to show that the sum of two short sides is going to be greater than the third side, Any two short sides or any two of the sides will be greater than the third side. So we need to show that our s plus S t is greater than rt We need to show that are s plus r T is greater than S t and finally s t plus R T needs to be greater than our s. Let&#x27;s go ahead and substitute our values for our inequalities. Rs the squared of 272 plus S t the square root of 10 88 is that greater than the square root of 2 72? Well, I don&#x27;t even have to go to the calculator for this one because we know that the squared of 10 88 is greater than zero already. And if we were to subtract the score 2 to 72 from both sides, that&#x27;s what we would have. So we can conclude this is a true statement. Moving on to Rs plus R T is the square root of 272 plus the square root of 272 is that greater than the square root of 10 88. This one will go over to her calculator squared of 2 72 plus square to 72. So we see it&#x27;s approximately 32.98 Let&#x27;s compare that with the square root of 10. 88. 32.98 What do you notice here? Both of these are equivalent. So we cannot conclude that the some of these two sides is greater than the third side because they are equal. And because they&#x27;re equal, it does not satisfy the triangle inequality through, um so we cannot form a triangle with these three vergis ease.

Steven Swee · Answer

It&#x27;s a real problem we&#x27;re given and a few things, and we don&#x27;t find the coordinates of tea. So this may sound scary at first, but let&#x27;s walk through it on one step at So in a problem statement we&#x27;re giving the triangle TV is a 30 60 90 triangle. That&#x27;s important. So we relate the silently together. Um, we know that you is a 0.3 negative. One three is at *** three negative one U Visa Hot news and teasing quadrant. What? There&#x27;s a lot of things to process. So let&#x27;s But let&#x27;s begin. Let&#x27;s know what planning you and B. So, in terms of approximating the points here, um, you would probably be around here. I have three. I say, um, we want you to be a three native one, so you would probably be around here. We will be a ***. Three negative one. Come here. And since this is a 30 60 90 triangle here, um, and teasing quarter of one, then we probably want to put t somewhere up here. Okay? Such that we have this triangle here. Okay, so that try and grow up here and it could be up here. Um, as long as teasing Quantrill one. And that should be okay. We just know that UV is the length of the iPod Minister. Now, how do we reconcile this? This is just basically not skilled drawing and drew it to skill that you would probably you might find the cornet City, but let&#x27;s say meow. Well, not that precise without instruments or without drawings. And we can&#x27;t figure out the corn&#x27;s the teachers were drawing it. So how do we use we recognize a UV? Here is Thea Apartness and 30 60 90 triangle Here. Here we look at a 3 69 triangle. The relationship between sidelines here that&#x27;s opposite to the respective angles would be X, um, ex Route three and two ex. Okay, so we know that I&#x27;m the length of the high part nous than is to x here. Hey, I&#x27;m not to confuse it with the x axis. It&#x27;s two X, and that&#x27;s the length of U V. As A to X is equal to the length of UV there. And since you use that three things one and Visa *** three native one here only differences in their X coordinates. We can see in that life and U V is equal to six, right? I&#x27;m you could apply any distance formula or realize we&#x27;ll use that three native one here and V is on the same horizontal line here. And then you three native one. Since the white cordons are the same, then the distance is calculated by their difference in acts coordinates. So the difference between three and then you&#x27;re three or three minus nine year three. It&#x27;s six to realize that X is equal to three here. Okay, now, visually have acquaint our diagram, and it looks like tea might be closer to you than to fi, right? And what you find is that the answer? Yes, it eyes the case there. Um, and here&#x27;s another reason why I&#x27;m we know the tea&#x27;s call. CTU is that, well, t u v or equidistant from the white access. Okay, So because he had white access is one xs equal to zero and three *** three arm, uh, least in terms of absolute value, the same distance away from the white access right V is three units the left and you is three inches to the right and the teas and quanta, and one then we can see. Definitely that he is closer to you than his TV. Okay, so looking this triangle here, we know the length of that pot news here is six. That means right. We know that UT the shortest like, is giving a length of three. And TV is gonna be of length. Ex Sorry. On three or three extra. 3333 Since we saw her ex already. Okay, so that&#x27;s where things start to get tricky. Like, how the heck do we find T there? And for that, we need to set up a system of equations. So in that case, let&#x27;s assign the coordinates of tea to be some ex not why not bodies there. Okay, so it&#x27;s a sign that pardon city to be ex not. Why not? What was relation to you here? Right. We know this distance is three. And by using the distance for actually, let me do something strange. We know that t here. Um, use different colors. T here has the coordinates. X not. Why not? And you have the coordinates for this According to the problem statement three native one. And we know the length of U. T. Is equal to three. So, using the distance, Formula three is equal to the square root off. Let&#x27;s see Ex, not ministry squared. Plus, why not? Plus one squared. It&#x27;s plus one. Because we&#x27;re taking was a track lending of one There. Okay, Now we square both sides get nine is equal to x, not ministry squared. Plus, why not? Plus one sweat there. Okay, so therefore, nine is equal to x, not squared. Um, Minus six bucks. Not plus nine. Plus, why not squared Plus, um onesie plus two. Why not? Plus one there. And we get that ex not swear my six x not plus. Why not squared? Plus two. Why not? Is equal to a movement on the constancy. Other side. It&#x27;s the negative one. Very Okay. Hey. So we have this ugly looking for most. Not easy to solve for X not and why not? But let&#x27;s see you do something with, um TV and figure out what? Where the if things and simplify. So, looking at TV no, t is X not. And why not? And V here is through three. So I&#x27;m V is at number three, Native one and distance TV. Here it is three for three. OK, so using the distance formal through three is equal to the square root here of, um, that&#x27;s not plus three squared. Plus, why not? Plus one squared scramble sites. That&#x27;s three score times three or 27 is equal to X, not plus three squared. Plus, why not? Plus once word, actually, I should have done here, let me do a racing first. Is that this? Actually, there&#x27;s an easier way to somewhere. So you will do everything here. Is that my nose here? And this is a little bit outre trick is that we have the term. Why not plus one here. And why not? Plus one over here. So let&#x27;s software. Why not with one? So we know that why not? Plus one squared is equal to 27 minus X, not plus two requiring this work there, which we can substitute here into our previous equation. Okay, So based on what we have here, um, why not? Plus one squared meantime bread? Why not? Plus one squared is equal to 27 minus x not minus. So it plus Therese worked. Okay, so therefore we get nine. You&#x27;re a Hillary. Put that in like nine is equal to X, not mice. Three squared, plus 27 minus X not plus three square. Hopefully something&#x27;s simplify. That&#x27;s where all the constants of love site So get negative. 18 is equal to have a factor. So I get It&#x27;s not squared. Minus six x not plus nine minus the quantity of ex not squared on plus six x night. Um, here, plus nine. Here. Okay, so that would get us your native 18. Easy for you. I just have a little bit. Bring for you. It&#x27;s not square demise. Exxon Square zero, then six x not minus six. Sex, not here is neither. 12 x not plus nine minus nine. Get zero. So therefore, X night here is native 18 or negative 12 here. So 18 over 12 which then simplify. Here it&#x27;s a three were, too. Which is 1.5. Yes. Oh, that little pauses. Me thinking like, Oh, these terms are canceled. Zero turns out, doesn&#x27;t it? Him and goodness. So you&#x27;re excited to go to 1.5? Okay? And this seems assistant with TB in Cuba. Why me? A positive x file here. Now we just need a positive wide value for tea. And since we know that, why not? Plus one square is a good 27 mines to quantity. Excellent post three squares going over here since we know that it&#x27;s not is equal to 1.5 we get. Let&#x27;s, um, factor was one squared. You see what the 27 minus 1.5 plus three squared? Okay, so we can get, um Why not plus one squared. I&#x27;m gonna avoid factoring here because my factory us then I guess a quadratic formula wasn&#x27;t quite equation. So, like, why not? Plus two white nuts plus one is able to see constant with the constant bickering back form that seems a little bit messy. Whereas we do it this way. Here, 27 minus 4.5 words there and my permission. I calculated that because, um, there&#x27;s something that you could probably do my hands, but I want to be able to be too long. 27 minus five squared will get you 6.75 Why not? Plus one is equal to the square root of 6.75 there and we don&#x27;t know is a plus or minus there. Right, well and we&#x27;re for Why to be and nor for tea to be in quadrant one. We need wine to be positive. So we only the postman, as we just said, Why not people? A positive route. So therefore, thing why not is equal to square root of 6.75 Why this one? Okay, And your funding here that why not? Here is approximately equal to 1.6 as our last co ordinate, right? And the real number is 1.59876211 So that&#x27;s something you probably wouldn&#x27;t need a calculator for. So the corn, it&#x27;s a tea then would be based on our findings. We found that exercise 1.5, and why not to be 1.6 or approximately 1.6. So would sleep at that. Nothing to be the co ordinates for tea. So this is in my having a pretty tough problem just because a lot of moving parts and having the worker the cornices home on, um, with a right triangle. That&#x27;s not what that doesn&#x27;t have one foot sites that&#x27;s on the Y axis on the X axis. That makes you more challenging. So just to recap. The first thing we did was try to process on the information we know. Now Here s a plot The points unity realized to use you greased iPod news and teasing Kyu won. We looked were used information that trouble TV is a 360 90 triangle are blamed The site links there and knowing that you were his iPod news, right? He knows that, um, the cycling&#x27;s hear the silence of the short side. The surest sign. We should be three. Yeah, So we drop this little figure here were related. So we just gave co ordinates Exxon Whiner 40. And we can solve this system. Where? Equation using two different to distance formulas. Right? And my solving that system we got X not is equal to 1.5. And why not to be approximately 1.6? Bear my opinion, I think the hardest step would be figure out that you had to use this distance formulae here and solving the system of equations and the system equations there. Um, he didn&#x27;t realize that you could substitute your substitution with the Quanta. Why not? Plus one squared. And that&#x27;s lovely algebra to work with, But yeah, this So this is how the system simplifies our desist solution here. And that was the final answer

Jose Hannan · Answer

Hello, everybody. In this video, I&#x27;m gonna be showing you how to solve exercise 33 chapter 13 Section four of calculus early. Transcendent that this problem gives us three points a ordinate 000 Be with coordinates 301 and see with coordinates 110 And they tell us to find the area of the triangle. It is formed by these three Vergis ease, as is drawn here to do this. The first thing we want to notice is that this triangle represents an area that is half of a larger parallelogram, which is this in this area that we adhere to. The original triangle has the same area, which means that the area we&#x27;re looking for is half the total area of the entire Carol Grant. So essentially, what we want to do is find the area of this entire parent parallelogram. To do this, you want to find two vectors, a B which extends to here and also encompasses this top side of the parallelogram as well as a C. This vector also takes the right side of this parallelogram. Let&#x27;s first see what these vectors are. We have the A B Victor is equal three minus zero, which is three zero minus zero, which is zero and one minus zero, which is one. Likewise, the victor a C its components one minus zero is one one minus zero and zero minus zero. And now we can see that this is a parallelogram that is formed by these two vectors A B and A C. Now, if you recall from 13 from fear um, 13.3. The area of this parallelogram is gonna be the magnitude of the cross product of these two vectors and therefore the area we&#x27;re looking for There&#x27;s gonna be equal 1/2 of this magnitude to find this magnitude. Let&#x27;s first calculate the cross product of a B and A C to do this. What we want to dio is find a determinant of a three by three major which is formed by setting its top row to be the three standard unit vectors. Second row to be the components of a B free Syria would one and the last year to be the components of a C 110 To find this determinant, we first multiply. I buy the determinant of the sub metrics here on the bottom, right? Zero one 10 We didn&#x27;t subtract a similar quantity for J sub Matrix here. It&#x27;s formed by this bottom left sub column in this bottom right sub column makes three one one zero and then finally, we have the same kind of quantity for J for K. I mean, which is this bottom left sub matrix here, three, zero one and one. Next we compute the determinants of these 32 by two matrices. The first term we have zero time zero minus one times one His negative one gives us minus I. And then from that we subtract three times zero which is zero minus one times one which is one and therefore we add one times j Then for the last term, we add three times one which is three minus one time zero which is zero gives us three times. Okay. And therefore the cross product of a B and A C is a vector components negative. One one, three. So now that we have this cross product, we can determine its magnitude. Find this magnitude. All we need to dio is at its components in quad richer. So this manage it becomes the square root. Negative one squared plus one squared plus three squared. This is equal to the square root one, it&#x27;s one nine. And as one plus one plus nine equals 11 we have that. This magnitude is the square of 11. And now, finally, as we had before, we know that the area is gonna be half the area of this parallelogram. So all we need to do is take this magnitude we just found divided by two. And then our answer is that the area of the strangle is a square of 11 divided by two. And this is our answer. So that&#x27;s how you solve exercise 33.

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Use the terminology from Section 8.2 Repeat Exer…

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

Catherine R.

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

Linear Algebra and Its Applications