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Let" = Ji+k v =ai + 2aj+ak: w ="i+ak0 > [ The volume of the parallelepiped determined by the vectors uVw is0 - 0t ~47/240 2u2...

Question

Let" = Ji+k v =ai + 2aj+ak: w ="i+ak0 > [ The volume of the parallelepiped determined by the vectors uVw is0 - 0t ~47/240 2u2

Let " = Ji+k v =ai + 2aj+ak: w ="i+ak 0 > [ The volume of the parallelepiped determined by the vectors uVw is 0 - 0 t ~4 7/2 40 2u2



Answers

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 ).

Welcome back to another cross product video. We'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'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's get that out of here. Ed, we'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'll use those values instead. When we ignore our first column, We're looking at one times 1 And it'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'll ignore a third column and then we'll look at zero times one -1 times one zero times one, one times 1, all multiplied by zero. Since we don't have any eyes jay's or k's, this is just a scalar and so we can add it all up. We have zero times one minus negative one and it'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.

Fun before. So it is getting asked one. Why do you? You don't want one? 11 is equal still you know my ass wanting to minus one this deep roots to want unit you says dancer.

In this problem, we'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'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's gonna be zero minus the three times W two be three is one W two is four. So that's gonna be for and then, for these second number, we have V three times w one That's gonna be one time seven, which is just seven minus V one times w three w three is zero. So it'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'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's gonna be native 12th plus zero times seven. That's going to be zero and then plus negative four times native three. That'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'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

What about the problem? Spanned by the trumpeters, u V and W we equal to the determinant after three fighters you ve end up in, you hear? They were taking that. Absolutely. This question was even the better. You you go to you. No, 2 to 1. A bit of the ego to 103 The better it up in your ego, Jr zero minus 40 So from here, we can find a volume by taking the determinant off the 2 to 1103 And then we have the zero minus 40 Remember, that would be the absolute value here. Then we equaled you. Yeah. The trick here will spend along on the road that having the most zero here, therefore, the determinant gender. Uh, we spent about this one. So we have on who have the plus there minus plus and minus. So have a minus minus will be far times the determinant, the semantics to 113 And then we get echo to the four times this determinant equity the five or six minutes. When you go to five, you go to the 20


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