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Question

Find the volume of the parallelepiped determined by the position vectors $\mathbf{u}=\langle 3,1,0\rangle, \mathbf{v}=\langle 2,4,1\rangle,$ and $\mathbf{w}=\langle 1,1,5\rangle$ (see
Exercise 63 ).

Jose Hannan · Accepted Answer

Hello, everybody. In this video, I&#x27;m gonna be showed you had to solve Exercise 64 chapter 13 Section four of calculus early, Transcendental. And this problem were given a parallel a pipe head that is formed by the vectors U V and W where you has components 31 and zero. He has components to 41 and W has components 11 and five. And with this information that wants to find the volume of the parallel a pipe head. To do this, we&#x27;re gonna have to recall a few important results from previous exercises. Let&#x27;s start with the result of Exercise 63 which states that the volume of such a parallel a pipe head is equal to you dotted with across W. And so this is a quantity that we want to calculate. To help calculate this. We&#x27;re also going to use a result from exercise 62 which states that the quantity in the absolute value signs you v cross W is equal to the determinant of a three by three matrix. His top row has the components of you three 10 Who second row has components of E to for one and his last year, has the components of W 11 and five. So what we want to do is calculate this determinant and then take its absolute value in order to get the volume that we&#x27;re looking for. So let&#x27;s break down this determinant. This is gonna be equal to three times the determinant of the two by two sub matrix on the bottom. Right here for one one. It&#x27;s attracted from this. We&#x27;re gonna have one times the determinant of the sub matrix form by this bottom left sub column. In this bottom writes a column to one one in five and in the last term, zero times another constant. So we can neglect this term as it is multiplication by zero. So now let&#x27;s expend out these two determinants in the first term. We have the three in front times four times five, which is 20 minus one times one, which is one it&#x27;s attracted from this we have two times five, which is 10 minus one times one, which is one. And so now carrying out these operations, this first term becomes three times 20 minus one, which is 19 and subjected from that, we have 10 minus one, which is nine and now three times 1957. It&#x27;s attracting nine from 57. We get 48 and so to conclude we have that. The volume of the parallel a pipe head is equal to the absolute value of 48 which is just 48 as it is already positive. And so here we have the volume of the parallel a pipette, and that&#x27;s how you solve exercise 64.

Vikash Ranjan · Answer

we have to find the border Deputy will do more. Cannot be dropped. C This includes 2123 minus 112 to 4. This is given as one into four minus two. My student minus two or minus four. Last three to minus one minus Students is given to black 16 minus nine. This is people equal to nine. Unit. You This is dancer.

Rishi Kavikondala · Answer

in this problem, we&#x27;re given three vectors U V and W. And the first part is to find the scaler triple product, which we can do by using this formula listed at the top. So for part A, let&#x27;s start off by finding the cross product the cross w so ve crossed w is equal to Okay, Well, starting with the two times w three, so be, too is one W three is zero. That&#x27;s gonna be zero minus the three times W two be three is one W two is four. So that&#x27;s gonna be for and then, for these second number, we have V three times w one That&#x27;s gonna be one time seven, which is just seven minus V one times w three w three is zero. So it&#x27;s just going to be for the second part. And then for the third part of the cross product, we have V one times w two v one is one w to us four. That gives us four minus V two times w won&#x27;t be too is one Billy one is seven. So that gives us seven. And when we simplify this, we get that the cross product is equal to negative for comma seven comma native three. Now we have to find the dot product of this vector and you so you you got be cross. W is equal to these sums of the products of the corresponding terms in these two vectors. So starting off with the first terms in each of the two vectors, we have negative for times three. That&#x27;s gonna be native 12th plus zero times seven. That&#x27;s going to be zero and then plus negative four times native three. That&#x27;s just 12. And when we find the sum of these three terms, we get zero. And obviously the magnitude of zero is just zero. So therefore, our scaler triple product is equal to zero. Now for part B, we&#x27;re asked to determine if these three vectors are co plainer. We know that they are complainer because the scaler triple A product is equal to zero. And we also know that if the volume is zero, then these three vectors are co player. The scaler triple product is equivalent to the volume, which means that since both are equal to zero, these three vectors are co plainer and the volume is zero. So our final answer is that they are co plainer and the volume is zero

Dylan Bates · Answer

Welcome back to another cross product video. We&#x27;re going to try to calculate the volume of a parallel pipe bed defined by these three vectors here. The textbook gives us a nice formula that we can use which says that it&#x27;s the absolute value of a dotted with a vector secrecy. But instead of calculating across product and then adopt product, we can do this using a triple product where instead of using I J and K in our matrix. Let&#x27;s get that out of here. Ed, we&#x27;re going to directly plug in are vectors A. B and see So doing that. We get 1 I1J zero. Okay. zero I 1 J one K and one I one J one. Kay. Then we can calculate the cross products same as normal but anywhere we have an I. J and K. We&#x27;ll use those values instead. When we ignore our first column, We&#x27;re looking at one times 1 And it&#x27;s one times 1. One minus one times not I but rather no. Yes. Now we ignore our second column And we look at zero times 1 -1 times one zero times one -1 times one. All multiplied by one us. We&#x27;ll ignore a third column and then we&#x27;ll look at zero times one -1 times one zero times one, one times 1, all multiplied by zero. Since we don&#x27;t have any eyes jay&#x27;s or k&#x27;s, this is just a scalar and so we can add it all up. We have zero times one minus negative one and it&#x27;s one plus negative one time zero at zero plus one plus zero, Giving us a volume of one unit cubed. Using the triple product method. Thanks for watching.

Vikash Ranjan · Answer

fun before. So it is getting asked one. Why do you? You don&#x27;t want one? 11 is equal still you know my ass wanting to minus one this deep roots to want unit you says dancer.

Rishi Kavikondala · Answer

In this problem, we have to find the scaler triple product given vectors, u V and doubly. Let&#x27;s start off by finding the cross product to be cross w recross w starting from the beginning is equal to the two time still you three. So that&#x27;s gonna be zero times one just just zero minus, maybe three times W two. So that&#x27;s one times negative one. So that&#x27;s just negative. One comma V three times w one w one is zero. So that automatically becomes zero. And then v one times w three. So everyone is negative. One totally three is one. So that&#x27;s just gonna be negative. One comma, then we have V one times W two. That&#x27;s a negative one times negative one Just one minus the two times W what? That zero time zero just gives us zero. And now when we simplify this cross product, we get one comma, one comma. What? Now We have to find the dot product of you and the answer we obtained here. So to do that, we have to find this some of the products of the corresponding elements in these two directors. So you dot he crossed w is equal to one times one, which is just one less one times negative one, which is just a negative one one times zero, which is just zero. And this just becomes one minus one, which in the end is zero. Because the scaler triple product, even without finishing finding for the magnitude, has given a zero for this dot product. We know that the answer to part A is automatically zero because the magnitude of zero is simply zero and because we received an answer of zero for part A. We know that these three vectors are co plainer because the scaler triple product is actually equal to the volume of the shape we&#x27;re trying to find. And since this volume is zero, we can assume that they are complainer. And so the answer here is also zero.

Problem

Explain why the position vectors $\mathbf{u}, \ma…

Problem 64 Hard Difficulty

Find the volume of the parallelepiped determined by the position vectors $\mathbf{u}=\langle 3,1,0\rangle, \mathbf{v}=\langle 2,4,1\rangle,$ and $\mathbf{w}=\langle 1,1,5\rangle$ (see
Exercise 63 ).

Answer

The volume of the parallelepiped is $48$.

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The volume of the parallelepiped is $48$.

Calculus Early Transcendentals

Lily A.

Kayleah T.

Samuel H.

Joseph L.

Vectors Intro

Vector Basics - Example 1

William Briggs, Lyle Cochran, Bernard GilletCalculus Early Transcendentals

Lily A.

Kayleah T.

Samuel H.

Joseph L.

William Briggs, Lyle Cochran, Bernard Gillet

Calculus Early Transcendentals