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

(a)-6(b) 6

Calculus 3

Chapter 9

Vectors in Two and Three Dimensions

Section 5

The Cross Product

Vectors

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in this problem, we have to start off by finding the scaler triple product. Given the vector is UV and doubly so. Let'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's negative. One time zero, which is just zero minus V three times W two. That'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's native one times six. That'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'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'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

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