three-vectors-mathbfa-mathbfb-and-mathbfc-are-given-a-find-their-scalar-triple-product-mathbfa-cdo-6

Question

Three vectors $\mathbf{a}, \mathbf{b},$ and $\mathbf{c}$ are given. (a) Find their scalar triple product $\mathbf{a} \cdot(\mathbf{b} 	imes \mathbf{c}) .$ (b) Are the vectors coplanar? If not, find the volume of the parallelepiped that they determine.
$$
\mathbf{a}=2 \mathbf{i}-2 \mathbf{j}-3 \mathbf{k}, \quad \mathbf{b}=3 \mathbf{i}-\mathbf{j}-\mathbf{k}, \quad \mathbf{c}=6 \mathbf{i}
$$

Ryan Mcalister · Accepted Answer

this question wants us to find the scaler triple product of a of three vectors vectors that it gives our vector A which is equal to to I minus two J minus three k. I&#x27;m gonna rewrite this as just one vector. Right now, it&#x27;s a what&#x27;s called a linear combination of three different unit vectors, all scaled and added together. I&#x27;m gonna rewrite This is just one vector, and that&#x27;s going to be just a simple s too negative to negative three. Because as we know, I is 100 j 010 and k is 001 when we scale them and added together, this is what we get. I&#x27;ll do the same thing for the other vectors. B, which is, um three. I minus j minus K is just equal to three negative one negative one and see, which is just, Ah six. I is just going to be 600 So these are our vectors and were asked to find a scaler triple product. The scale a triple product is, of course, from your book a dotted with Be cross. See? No, let&#x27;s start with finding be cross e b crusty. We&#x27;re gonna find using our regular method of the determinant of a three by three Matrix will have the first row. B i j K are unit vectors. Our second and third row will be B and C. So three negative one negative one and of course, zero or excuse me six 00 When we find the determinant of this will do a co factor, decomposition along the first row will get the determinant of negative one negative 100 times I minus. Remember, we subtract when we&#x27;re doing the J term here. 36 negative 10 times j And then we add the determinant of 36 negative 10 times k. Now we know how to find determines of to a two. Major sees it. This is just going to be zero I, uh, minus six J because we have a positive six as their determinant and we are subtracting it and then we&#x27;re adding six case. Let me rewrite this and a little bit prettier of fashion is just gonna be negative. Six j plus six k. And in other words, that is zero negative. Six positive six. So we have our cross product now all that&#x27;s left to do is dot it with our a vector, remember, Are a vector was too negative to negative three So too negative to negative. Three dotted with zero negative. Six negative positive. Six. And what we do here is we take our, uh we add up the products of the corresponding, uh, components. So are I. Components multiplied together is to time zero, which is zero r j components multiple together. Negative two times. Think of six is positive 12 and R K components negative. Three times six isn&#x27;t negative. 18. So what that is 12 miners 18 is, of course, negative six. And that is our scaler triple product a dotted with the across C. Now the next part of this question asks us if this is co plainer. And if not, it wants us to find the volume of a shape that these three, uh, factors make what we know right away that since the scale a triple product is negative six. It&#x27;s not complainer. Any skeletal product that&#x27;s not zero is from three factors that are not co plainer. So a and B and C or not complaining, which means they are not all in the same plane, so they formed a parallel, a pipette apparent little pipe ed. If I can try to draw it on a two dimensional screen, will look something like this. Let&#x27;s say this is B and C is somewhere behind it. So let&#x27;s imagine this is ah, a little bit behind B. And there&#x27;s an angle here and this is in one plane. Of course, any two vectors, um, can always be put in tow. One plane and now a we know is not in this plan, so it&#x27;ll be sticking out a little bit. Let&#x27;s say it&#x27;s up here. The parallel of hype Ed, this defines is something like this. It has a base of B and C that will look like this, let&#x27;s say and then height defined by a like this, and we can connect this all here, and it&#x27;s not very well drawn, but you can imagine what it really would look like. So it&#x27;s essentially a parallelogram prism, but then you slanted a little bit with a This is Specter A So how are we gonna find the volume of this? Well, there&#x27;s a trick, but I want to show you the long way first, and we&#x27;ll get, um we&#x27;ll meet up with the trick at the very end, and it&#x27;s pretty exciting. So, uh, buckle in first, we want to find for any prism. We&#x27;re always gonna find the volume as base times height. So let&#x27;s find the volume of the base. First, we&#x27;re gonna treat B and C as the base. So here is vector B, and this is now the two dimensional, like looking straight at the base from above. And here, see? And we&#x27;ve got, like, something like this and something like this. So when we&#x27;re looking at B and C, let&#x27;s say this is data DiVall In the area of a parallelogram like this is the base times the height perpendicular to that base of this right here. Well, we know what see is and we know if it is. So this height right here is simply see sign Seda, Remember, since we&#x27;re finding the opposite side of this right triangle here that&#x27;s gonna be see Scient. Ada. So the height or excuse me? The area of this parallelogram is the magnitude of B times, the magnitude of C times sign of data. And now this is where it gets exciting because you probably already recognize this as the magnitude of be cross. See? So the magnitude to be cross see is what we&#x27;re looking for as the base. Well, we had we be cross. He was part of this scale, a triple product. So let&#x27;s see how this will play into it. We have the base and we want the height. While the height again is not gonna be a it&#x27;s going to be the, uh, the portion of a the component of a that is perpendicular to our base or orthogonal from our base. What we know that be cross see, is orthogonal to this Plan B is B cross. He is perpendicular to be and to see. So if there&#x27;s an angle fader between be cross C and A, we want to find the component of A that is alongside this, uh, this value here alongside Be cross, See? Well, that&#x27;s gonna be a times this co sign of data. The blue part is eight times the co sign of Fada. So we want to multiply the base times this height. The base we know is the magnitude of be cross. See, I&#x27;m gonna write this in. Ah, let me to read here. The base is a magnitude of be crossed, see? And the height is the manager of a times the co sign of Fada. And now this is where it gets really, really exciting. Because this is exactly the formula for a dotted with be cross. See? Well, hold on. We just figured out what a daughter be cross. He was. We said it was, um we said it was negative. Six. So we&#x27;re finding the magnitude of negative six. Well, magnitude just means absolute value. The absolute value of six is, of course, positive. Six. So we found that the volume of this scale a triple proud of this peril pipette is equal to the absolute value of the scale. A triple product, which is six. And that&#x27;s gonna be true for any three non co plainer, uh, vectors. The area of the volume of this parallel pipe. It will be the absolute value of the scale. A triple product. So this is the really exciting part. Um, I hope you enjoyed. That&#x27;s your final answer.

Ryan Mcalister · Answer

this question gives us three vectors and asks us to find thes scaler triple product the vectors that it gives our a vector A is equal to I minus j plus k. Remember, I is just 100 j 010 and K is 001 So this is really a linear combination or a some of unit vectors When we add them all up together, What we actually get is just one negative 11 So this is how I&#x27;ll be writing. All of our vectors that were given in this problem is going to make it easier to deal with as we move on through finding this scale. A triple product vector B is minus j plus K or zero negative +11 and vector See, is I plus J Class K or +111 So here are vectors and what we&#x27;re finding is that the scale of triple product that is a dotted with be cross see since be crossed seas and the parentheses. We can do that first. In fact, we have to do that first by order of operations. So let&#x27;s jump into it. Be cross See will be found using our standard method of a three by three Matrix whose first row is I. J K are unit vectors and second and third rose will be B and C. So are second rate will be zero negative 11 and our throat third world will be 111 and we&#x27;ll find the determinant of this. And this will give us a vector that is the cross product of B and C. So we&#x27;ll do this using co factor decomposition along the first row here. So we&#x27;ll have, um, our first co factor, which is negative. 111 And one times I will subtract the next one because of where J is in the Matrix. So we&#x27;ll subtract the determinant of 011 and one times J. Then we&#x27;ll add our third term, which is zero negative 11 and one times k. So we have a couple two by two matrices. By now we&#x27;re experts at finding determinants of two by two matrices, so we&#x27;ll find that this is going to be negative to I plus J. It&#x27;s a negative one j but we&#x27;re subtracting it. So it&#x27;s plus J and then plus K. So this is what we have. We have negative two I plus J plus K and turning that into just one factor. What we&#x27;ll have is negative 21 and one. So that&#x27;s our cross product. That is be Cross E. And now all that&#x27;s left to do is dotted with a So a dot be cross C is going to give us is going to be one negative one, not one dotted with negative 211 just to review dot products that products are the some of the products of thecornerscores ending components. So what we&#x27;ll do is we&#x27;ll multiply our I component. So one times negative two is negative. Two. We&#x27;ll add the product of her J components negative one times one, which is negative one. And we&#x27;ll add the product of our kidney components one times one, which is one so negative two plus negative one plus one is equal to negative two. So that is our scaler triple product. Negative two is a dotted with be across C. So the next part of this question asks, Are these vectors co plainer? And if they&#x27;re not, what is the volume of the parallel? A pipette that they define well we know right away that they&#x27;re not complainer. And we know that because if they were co planner, the scale of triple product, it would be equal to zero. And since it&#x27;s not, we know that these air not co plainer. But what is the volume of this parallel? A pipette that they define a parallel a pipe ed would be is a little tricky to draw on a two dimensional surface. But imagine that you have a vector. Be here vector. See here and there in a plane. So seize a little bit behind Be evil Spare with me on this And then we have vector a coming out like this going up and away. However you might imagine it the parallel pipe. It uses B and C as the base. So imagine this is the base of parallelogram and then each thes corners goes up to define the height. We just connect thes all. It&#x27;s like if we had a parallelogram and made it into a prism and then slanted that prison a little bit. We have this. So what we&#x27;re looking for is the volume of this shape, which is going to be obviously the base times the height. So let&#x27;s start by finding the volume of the base. Well, before we do that, let&#x27;s I&#x27;m gonna draw it two dimensional. This is just the base. Here&#x27;s being here, See? And what are we looking for? Well, we&#x27;re looking for be times this height here. Well, this height here, we know, is if this is the angle theta, we know that this height there is just gonna be c sine theta. So really, what we&#x27;re looking for is the magnitude of B times, the magnitude of C times, the sine of the angle and between and hold on. We should recognize this formula right away that is equal to the magnitude of be cross, see, so that we already know we can already figure out what that is. But we have another trick that we&#x27;ll see in a second here with a that&#x27;s gonna make this even easier. Would be cross, see is if I draw it in our first diagram is something like this. So it&#x27;s strict. It&#x27;s six straight up out of the b c plain, and that is gonna help us a lot, because when we find the height, we&#x27;re not finding the length of A We really want to find the height of this parallel a pipette that is perpendicular to the base that is straight up out of be cross, See So we want to find the length of a along Be cross, See? Well, if we say that this angle here between A and B cross sees data, then this hide that I wrote wrote in Blue is just a times the co sign of Fada So we&#x27;re looking for is be cross e the magnitude of be cross see times the magnitude of a times the co sign of data. And now hold on. You guys might be jumping up and down because this is also ah, formula that we know that is just going to be the magnitude of a dotted with e cross. See, And that is something that we just found a daughter with Be cross see, otherwise known as the scale of triple product was negative two. So the magnitude of this negative two is just positive two and we have found the volume of this parallel a pipe head. It&#x27;s just too. And that&#x27;s gonna be always be true no matter what our scale a triple product is. It&#x27;s always gonna be the volume of this parallel, a pipette that three vectors described, and so that is your final answer.

Megan Mcfarland · Answer

in this problem, you&#x27;re given three vectors and asked if I&#x27;m the dog product between you and the vector from the cross product to be cross W. And so first, we&#x27;re going to convert all of these doctors into the normal vector formula used to seeing Yume is going to be too negative too negative. Three b is going to be three negative one negative warm and w is going to be six 00 Okay, now we&#x27;re going to take the cross W this is per day be cross w, and that is equal to them Matrix Where the first rose i j k the second round with the components of B which are three negative one negative one the third row with components of W 600 Now we&#x27;re going to do this and we find that for eyes component to find that we&#x27;re gonna cross up the first column and take the determinant of this two by two matrix and detonated 1 p.m. zero comes or minus negative one times a year. That serum in a zero that&#x27;s just zero. Then we always attracted jeez, component. And to get Dr Boehner, we&#x27;re gonna cross at the middle call and take the determines of this to buy shoe matrix and we get three times zero minus negative. One time. 60 minus negative. Six year post 626 And to get Caisse component been across at the third Column, take the determines of this two by two matrix and we get three times zero minus negative. One time. Six That zero minus negative. 60 plus six is six and we have found our vector V cross W which is zero negative. 66 Now we&#x27;re going to take the dot part of this with the vector you, which again is to negative to negative three. Let me do this. We get the following two times zero Yes, negative. Two times negative. Six plus negative. Three times six Do time. 00 negative. Two times in a 6 to 12 negative. Three times six is 18 me a 12 minus 18. That&#x27;s negative. Six. So instead of part is going to be negative six. And then for Part B. They first asked us if the vectors or co planner determine if they&#x27;re co plainer. We see if the doc products between you and the cross product be crossed. W&#x27;s equal to zero in part a. We see that is not equal to zero. So therefore they&#x27;re not co plainer and then to find the volume of the apparel pipe ed. It is actually quite simple. All we do is take the dot product we found between you and the cross product of across W, and we take the absolute value of it. And so we&#x27;re going to take the absolute value of negative six. Let me see that that is equal to six, and that is the volume of a parallel piper.

Ryan Mcalister · Answer

this question starts by asking us to find a, uh, scaler triple product of three vectors. The vectors that it gives are A which is equal to 123 be, which is equal to negative 32 one and see which is equal to 08 and 10. Now the scale a triple product is going to be a dot be cross sea and order of operations will still apply here. We&#x27;re going to do the parentheses first, be cross, see and then take the dot product with a So let&#x27;s jump into it the same way we always do with cross products when you use a three by three matrix, the first row of which will be our unit vectors I J. K. And the second and third row will be the vectors himself. So negative. 321 and 08 and 10. When we take the determine of this three by three matrix, we&#x27;re going to dio co factor decomposition along the first row. That means we&#x27;re going to take ah minor 11 which is just the matrix without Row one and column one that is 281 and 10. The bottom right corner multiply that by I. Then we&#x27;ll subtract. Remember, we&#x27;re subtracting this next term because of where it is in the Matrix. Since J is Element 12 and one plus two is three, which is an odd number. We subtract this term this co factor and that co factors gonna be found by taking the determinant of minor one to which again is the Matrix without Row one and column to so negative 301 and 10 times element one to which, of course, is J. And then we&#x27;ll finally add the co factor. 13 which is the determinant of minor 13 negative 30 to 8 times the unit vector K. So if we do all of this really quickly, what we&#x27;ll find is we have 20 minus eight, which is 12 I minus negative 30 which is positive 30 j and minus 24 plus zero. So native 24 minus 24 k So this is our cross product. We have 12 i 30 j 24 k We can turn that into a vector. Just 12. 30 negative 24. And now what we want to do is take the doubt product with a So the dot product a dot be cross C is going to be equal to 123 dot 12. 30 is the negative negative 24 in the dot product. Just to review is we multiply each component and then add them together. So what we&#x27;re gonna have is it&#x27;s not a vector, it&#x27;s a scaler quantity. We&#x27;re gonna have one times 12 plus two times 30 plus three times negative 24 So will be left with 12 plus 60 60 60 minus 72 which is equal to zero. So our scaler triple product is equal to zero. As we found part A and part B asked us, Are these all co plainer? Well, when we have a scale trip of product equal to zero, we know that yes, they are co planner. Co planner means that all three of these vectors can be contained within the same plane. And I&#x27;ll, uh, give a little bit of explanation Is too. How we know this. Yes. So, yes, they are all cope later. So how do we know that skeletal product of zero means they are complaining? Well, let&#x27;s take a look when we found be cross see first we said, for example, here&#x27;s B, here&#x27;s C and we found Be crust. See here and be crust see with equal to or perpendicular to B and C Let&#x27;s be cross e. So no matter of what, When we have two vectors, we can always put a plane around them and I&#x27;m gonna draw that. Now let&#x27;s say we have a plane that contains B and C, and it looks something like this. Um, this is you have to believe me. Drawing in three dimensions is not easy. Um, so let&#x27;s just say that&#x27;s the plane that contains B and C. And now we know that be cross, see is perpendicular is orthogonal to that plane, so it&#x27;s sticking straight out. What we did is we found that a dot beak Rossi is equal to zero. Well, we know that any any two vectors let&#x27;s call them you and v u dot v equals zero means that you is perpendicular to V. So wouldn&#x27;t we found that? Be cross si dot a is equal to zero. We found that a is perpendicular to be across C. That means that a is perpendicular to this vector that we drew here. And the only way to be perpendicular to it is to fall into this plane that we just drew around B and C. So since a is perpendicular to be cross, see a is complainer with B and C, and that is how we know that all three of these factors are indeed cope later. And that&#x27;s your final answer.

Rishi Kavikondala · Answer

in this problem, we have to start off by finding the scaler triple product. Given the vector is UV and doubly so. Let&#x27;s start off by finding V Cross W The cross W which we confined using the formula listed here is V two times W three. So that&#x27;s negative. One time zero, which is just zero minus V three times W two. That&#x27;s gonna be negative. One time zero, which was just zero and second part V. Three times W one. The three is negative one. W one is six that makes this term negative. Six and then v one times W three Deli 30 Therefore, the second term is zero. And for the third part V one times W two w 20 So this can automatically be made zero minus V two times w one that&#x27;s native one times six. That&#x27;s just negative. Six. And now when we simplify this, we get an answer of zero common negative six comma six. Okay. And from here we can find the dot product of you and our cross product. I do. God, he crossed w The way we do this is by multiplying all the corresponding components together So starting from the beginning, we have zero times two, obviously just zero plus negative. Two times native six, which is just negative. 12. We&#x27;re sorry. 12 plus negative. Three times six, which is just negative. 18. So negative. Eat. That makes this answer equal to native six. Because we&#x27;re just doing 12 minus 18. So our answer right here is negative six. And because we did not get an answer of zero, this means that these vectors are not co plainer. In other words, they do not all exist in the same plane. We know that to find the volume. Therefore, all we have to do is take the absolute value over a scaler triple product. So the volume is just the absolute value of negative six, which is just six

Megan Mcfarland · Answer

in this problem were given three vectors and asked to provide the dot product between you and the cross product of V Cross W. And so first up in this is going to be to find that cross product to be across W. And so to do that, we&#x27;re going to first draw Matrix with its first row. Was I Jane K. It&#x27;s second row as the components of be because we&#x27;re doing be cross W. And so if we rewrite B and W in their normal doctor forms, we&#x27;re going to see the B is equal to the following victor. There&#x27;s no I components. We get zero. The J coefficient is negative. One in the K coefficient is one, and then W is going to be equal to the vector 111 And so these components go in the second row, and W&#x27;s components go in the third row. And so when we do this, we get the following eyes component to get that everyone across that the first call on take they call the determinant of this two by two matrix. It&#x27;s going to be negative one times one minus one times one or negative one of minus one, which is negative. Two. That we always subtract the Jays component and to get that warm going across that the middle column Take the determines of this two by two matrix and get zero times one minus one times warm zero minus one negative one. The negatives. Cancel that, and to get Kay when across at the third calm. Take the determines of this two by two matrix we get zero times one minus negative, one times warm or zero minus negative, warm, dizzy or plus one which is just one. And so we found our vector for the cross product of E Cross. W is going to be negative, too. One word. Everyone who doesn&#x27;t collect that with you and if we rewrite you into are the normal doctor. Former use of scene is going to be one negative 11 And so when we do this stop product, we get the following. We&#x27;re going to get war times negative, too, plus negative warrant times one close. The one times one one times negative two is negative. Two close later, one times one that&#x27;s negative. Once we get negative. Two minus one. That&#x27;s negative. Three and two that one times more was just wondering a negative three plus one. That&#x27;s negative too. And so our answer to party is going to be too negative too. Now, in part B were asked if the vectors air co planing to determine this. We have to see if the dot problems between you in the new Doctor V Cross W is equal to zero. If it is equal to zero, the vectors are complainer. We can see in this case it is not equal to zero. Therefore, the vectors air, not complainer. And so now we have to find the volume of the pair a little pipe ed to do. This is actually quite simple. We&#x27;re going to take the dog product we just found in part A, and we want to take the absolute value of it. So we&#x27;re gonna take the absolute value of negative, too, and we get to and that right there is the volume of the parallel Piper

Problem

Volume of a Fish Tank A fish tank in an avant-gar…

Problem 34 Hard Difficulty

Answer

the vectors are not coplanar. The volume of the parallelepiped is 6 cubic units, since volume cannot be negative.

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the vectors are not coplanar. The volume of the parallelepiped is 6 cubic units, since volume cannot be negative.

Algebra and Trigonometry

Catherine R.

Heather Z.

Kristen K.

Michael J.

Vectors Intro

Vector Basics - Example 1

James Stewart, Lothar Redlin, Saleem WatsonAlgebra and Trigonometry

Catherine R.

Heather Z.

Kristen K.

Michael J.

James Stewart, Lothar Redlin, Saleem Watson

Algebra and Trigonometry