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

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

Calculus 3

Chapter 10

Vector in Two and Three Dimensions

Section 5

The Cross Product

Vectors

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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'm gonna rewrite this as just one vector. Right now, it's a what's called a linear combination of three different unit vectors, all scaled and added together. I'm gonna rewrite This is just one vector, and that'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'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's start with finding be cross e b crusty. We'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'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'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'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'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's not complainer. Any skeletal product that'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's say this is B and C is somewhere behind it. So let's imagine this is ah, a little bit behind B. And there'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'll be sticking out a little bit. Let's say it'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's say and then height defined by a like this, and we can connect this all here, and it's not very well drawn, but you can imagine what it really would look like. So it'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's a trick, but I want to show you the long way first, and we'll get, um we'll meet up with the trick at the very end, and it's pretty exciting. So, uh, buckle in first, we want to find for any prism. We're always gonna find the volume as base times height. So let's find the volume of the base. First, we'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've got, like, something like this and something like this. So when we're looking at B and C, let'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're finding the opposite side of this right triangle here that'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're looking for as the base. Well, we had we be cross. He was part of this scale, a triple product. So let'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'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'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'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'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'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'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's your final answer.

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