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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.$$\mathbf{a}=\langle 1,2,3\rangle, \quad \mathbf{b}=\langle- 3,2,1\rangle, \quad \mathbf{c}=\langle 0,8,10\rangle$$

$\vec{a} \cdot(\vec{b} \times \vec{c})=0$$\vec{a}, \vec{b}, \vec{c}$ are coplaner.

Calculus 3

Chapter 10

Vector in Two and Three Dimensions

Section 5

The Cross Product

Vectors

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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're going to do the parentheses first, be cross, see and then take the dot product with a So let'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're going to dio co factor decomposition along the first row. That means we'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'll subtract. Remember, we'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'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'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're gonna have is it's not a vector, it's a scaler quantity. We'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'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's take a look when we found be cross see first we said, for example, here's B, here's C and we found Be crust. See here and be crust see with equal to or perpendicular to B and C Let's be cross e. So no matter of what, When we have two vectors, we can always put a plane around them and I'm gonna draw that. Now let'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's just say that'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'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's call them you and v u dot v equals zero means that you is perpendicular to V. So wouldn'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's your final answer.

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