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Volume In Exercises 37 and38, tind the volume of" the parallelepiped with the given vertices.37. (0,0,(), (3.0,0), (0,5,1). (2,0,5), (3,5, 1), (5,0,5), (2,5,6)...

Question

Volume In Exercises 37 and38, tind the volume of" the parallelepiped with the given vertices.37. (0,0,(), (3.0,0), (0,5,1). (2,0,5), (3,5, 1), (5,0,5), (2,5,6) , (5,5,6) 38. (0, 0, (), (0,4,0),(-3,0,0), (-1,1,5), (-3,4,0),(-1,5.5),(-4,1,5). (-4,5,5)

Volume In Exercises 37 and38, tind the volume of" the parallelepiped with the given vertices. 37. (0,0,(), (3.0,0), (0,5,1). (2,0,5), (3,5, 1), (5,0,5), (2,5,6) , (5,5,6) 38. (0, 0, (), (0,4,0),(-3,0,0), (-1,1,5), (-3,4,0),(-1,5.5),(-4,1,5). (-4,5,5)



Answers

Find the volume of the parallelepiped with the given vertices. $$\begin{aligned}&(0,0,0),(3,0,0),(0,5,1),(2,0,5)\\&(3,5,1),(5,0,5),(2,5,6),(5,5,6)\end{aligned}$$

Hi. So in this video I'm going to show how to find the volume of a parallel pipe depending on what we are given. So in this case we are given all of the coordinates. And so I just went ahead and graft it in a geology brah and you can just google that and I find it very easy for me to use it To Graph three space problems because then I can visualize it a little bit better and in this case actually needed to because then I could find three adjacent edges to use. So that's why I picked the highlighted three points. And those are going to be our three points that we're going to use in order to do our volume and to find volume, we know that we can use our dot product and our cross product and it's going to look like this. And we're going to do are cross product of V N W. And then we're going to do the dot product of argue. So let's first start out with doing our cross product and I'm going to set it up how I always do with a bracket and then I J K and then putting in the respective numbers and then I will go ahead and do the cross product and I'm going to do each one in different colours. Just, we can identify it a little bit better. So I is going to be 0105 and that's going to be an orange. And then let's do uh j in red and that's gonna be negative three negative 105. And that's J. And I don't know why I changed to zero, but hey, ipad things and then I'm going to use I think yellow for K. And it's going to be negative three negative 10 and one for K. Now, once you do all of the cross multiplying in the math, you can find out that it's going to be zero, I minus actually plus 15 J minus three K. And I'm going to just write that as a as a vector and it's going to be 0 15 negative three. Now I'm going to use this Cross products that we just found and use the dot product using the U. Values we have. So I'm just going to take zero 40 Times are cross product of 0, 15, -3 and I find that it's going to be zero. Yeah. Oh, so sorry, yep, it is going to be zero plus 60 plus zero. And then when you add it all together, you get 60 and this is going to be your volume of your object. And I hope that this helped a little bit more than what you guys had and that you guys understand our cross product and that product rules a little bit better

Hello. Hope you're doing well. So we're given these eight points here that formed the Vergis ease of a parallel pipe. We need to find the volume of this parallel pipe. So the formula for that for the volume is equal to the absolute value of the triple scaler product, which is you dot v cross w. So to find this triple scaler product, you could find V cross w and then find the dot product of you and that correspondent vector. There's an easier way to do that. You could just set up at the determinant of a three by three matrix with the first road containing the components of Are you vector Second road contains the components of our view Vector. The third road contains the components of R W back and then the way you find the determinant of this three by three matrix is you take your you one and multiply it by the determinant of this two by two matrix, then subtract you to multiplied by the determinant of this two by two matrix. Then add you three multiplied by the determinant of this two by two matrix that gives you you triple scaler product Now let's go and dive into this problem. So the first thing we need to do is we need to find our u V and W vectors, which are going to be three adjacent. Vector is that formed three adjacent sides of, uh, our three adjacent edges of the parallel a pipe. So it's kind of nice. This problem has points that essentially in kind of two different planes. So, uh, you have four points that have X values of zero and four points that have X values of four. So you can kind of think of our parallel pipette had split up between the points split up between two planes. One plane at X is equal to zero, and one Planet X is equal to four. So for you dot v cross, W R U and V are going into be into the going to be in the same plain. And then w will be like coming out towards us as our cross product. So r w vector. So let's take a as our starting 0.0 So all three vectors will come from that point so you vector or w vector will be coming out of the plane. So let's make that R W vector go from A to B vector A. B because it's starting at her point and kind of coming directly out of this X is equal to zero point. So to find the vector a B, we're going to take the individual values of the components of be subtract from that. The corresponding components from that means w is going to be equal to X component is going before minus zero, which is for why component zero minus zero, which is zero. And as e component is again zero minus zero, which is zero. This is our W factor. So now to find you are U and V vectors. We need to figure out what other two points that a connects with in its X is equal to zero plane. So let's draw X equal to zero plane and see what points point A is connected with. So let's graph are four points that are in the X is equal to zero point. So for 0.0.8000 let's point a right there. Let's see Point d A. Zero minus 23 It's gonna be somewhere like there. See point at 053 it's gonna be zero 53 And then point G is 036 It's gonna be somewhere up here. So this is essentially what our parallelogram and the X is equal to zero plane is going to look like. So as you can see a connects with D and F, So we need to figure out what our A D and A F vectors are. So let's make our let me see. So let's make our a d vector equal to V. Say a D is equal to R V. Vector is equal to on Ben. It's just going to we're going to take all the components of RD point and subtract the corresponding components on our A point. So because A is just 000 subtracting that gives us the same vector with the same components as they're deep pointy. So it's essentially zero minus 23 All right, so now moving onto a f let's make that are you vector because we have u V. And then we've got w coming out of the plane. Um, so let's make, uh, are you vector is going to be equal to the components in our F point minus the components are eight point and again because is just 000 are vegetable out. The same components is our point at 2053 Okay, so now we can we know RV and you vectors that we've got our w vector is equal to 400 So let's find the triple scaler product. So you thought the cross w equal to we're going to take the determinant of our three by three matrix. Our first row is going to be the components of Are you vector, which is 053 Second row is gonna be the components of RV Vector, which is zero minus 23 Our third row is going to be the components of R W Beck, which is 400 Okay, so we have, um so we're gonna take Are you? Oh, I'm sorry. Zero multiplied by the determinant. This matrix right here, which is minus 2300 and with these vertical bars on either side of the matrix means we're taking the determinant of the matrix. Got minus five here multiplied by the determinant of this matrix there, 0340 got plus three here, multiplied by the determinant of this matrix here, zero minus two for zero. So we need to review real quick how to take the determinants of these two by two matrices. So if we have a two by two matrix with components A, B, C and D determine, it will be equal to a D minus B. C. So keeping that in mind, we can go and simplify this. So we got zero times actually don't need to, because this is just gonna almost play out to zero, so you can not worry about that. So then we have minus five times a d a zero time zero, which is zero minus B c. That's four times three, which is 12. Got plus three multiplied by a D that zero time 00 minus B c. That's four times minus two, which is minus eight. Okay, so we've got minus five times minus 12. That's 60 zero minus negative. Eight. That's plus eight. Three times eight. It's plus 24. That's equal to 84. So remember the volume of our parallel parallel pipette is equal toe absolute value of you dot v cross w. That's equal to the absolute value of 84 which is just 84. Some means your volume is equal to 84. This right here is our final answer. This is the volume of our parallel A pet. All right. Well, thanks. And I hope that helps.

We are given. The Vertex is of a parallel a pipe head and we're asked to find its volume. The verses are A with coordinates 000 B with coordinates 111 Sorry. 110 See with coordinates 102 D with coordinates 011 E with coordinates 212 F with coordinates 113 G with coordinates 1 to 1 and H with coordinates 2 to 3. To find the volume of this parallel pipe ed, let's first find three vectors, which span it from the same point. So I'm going to plot each of these vortices, so I'll let X range from zero up to two. Let y range from zero up to two, and I'll let Z range from zero up to three. Okay, so a has coordinates. 00 zero. It's located about here. Be as coordinates 110 It's located about here. See? Has coordinates 102 It's located about here. Diaz coordinates 011 It's located about here. Yes. Coordinates 212 It's located about here. F has coordinates 113 It's located about here. GPS coordinates 1 to 1. It's located about here. NHS coordinates 2 to 3. It's located about here, so it's a little bit hard to tell, but can pretty clearly tell that E and H should be connected. I draw this connection in green towards the front. See, then it probably makes sense for each be connected to C C to be connected to F f to be connected to H. So we already have one plane there. Excuse me. Also for you to be connected to B B to be connected to G and G to be connected to H and finally G connected to D. D. Connected to F um d connected to a A connected to be and a connected to see So you can kind of see this box shape, which is our parallel pipe head. Maybe not the most accurately drawn figure, but does the job and we can see starting at the origin or point A. We find three vectors which span this parallel of iPad. So I'll call this one you. This one the and this one w and three vectors u V and W. They have component forms. Well, you is simply the vector from A to D, which has components 011 The is the vector from A to B, which has components 110 and W is the vector from eight to see which has components. 102 Okay. Now, to find the volume of this parallel a pipe head, we need to find the absolute value of the signed volume, which we know is the same as the Scalar triple product. You dotted with the cross with W. So this is equal to the absolute value of U dotted with the determinant of the matrix. I j Okay, 110102 This is the absolute value of U dotted with. Well, instead of you, I'll write out the whole vector 011 started with See, this is to minus zero or two minus two, minus here or negative too. And finally zero minus one or negative one. This is the absolute value of some of the products of components which is zero times two plus one times negative. Two plus one times negative one. This is the absolute value of zero minus two. Minus one or negative three, which is three. And because this is volume, this is in cubic units

Were given advertises of a parallel a pipe head. We're asked to find its volume. The verses are a with coordinates 000 B with coordinates 400 See with coordinates. Four Negative, 23 D with coordinates. Zero negative 23 E with coordinates 453 F with coordinates 053 g with coordinates 036 and H with coordinates 436 So, even though we have these vortices, it's not exactly clear how these vortices come together to form the parallel pipe head. So let's sketch a graph of the shape first and then we'll find its volume. So I'm going to let let's see the X axis range from zero up to positive four well at the Y axis range from Let's see, negative two up to positive five in the Z axis from zero up to six. Mhm. So we have a located here at the origin. Then we have to be located at 400 About here. Then we have C located at four. Negative 23 about here. Then we have d at zero negative 23 About here. Then we have e at 453 then we have f at 053 out here, then G at 036 and H at 436 out here. So the parallel a pipe head appears to look something like this. So I've drawn three of the spanning vectors and blue. I'll call these victors you b and W. Now it's clear that vector u So we started the origin. This has component form 05 three. Vector V has component form 400 and vector W has component form zero negative 23 The volume of the parallel a pipe head. Well, this is going to be the absolute value of the signed volume, which recalled assigned volume is the same as the scalar triple product you dotted with be crossed with W three vectors that span the parallel pipe head. So this is the absolute value of U dotted with the determinant of the matrix. I j k 400 zero negative 23 This is equal to the absolute value of 053 started with, So this is zero minus zero is zero than minus 12. Minus zero is negative 12 and finally, negative eight minus zero is negative eight. This is the absolute value of the some of the products of components, or zero times zero plus five times negative 12 plus three times negative eight. This is the absolute value of zero minus 60 minus 24 or minus 84 which is simply 84. And because this is volume, this is in cubic units.


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