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(15 points) Which of the following can be carried out? If it can be carried out, write what sort of object YOu get as an answer (scalar o vector). If they out cant ...

Question

(15 points) Which of the following can be carried out? If it can be carried out, write what sort of object YOu get as an answer (scalar o vector). If they out cant be carried briefly explain why not_ (To remove any doubt: are vectors; k is scalar constant, f is scalar function and F is vector function:) (u Vf div(kF) gradF curl(F)

(15 points) Which of the following can be carried out? If it can be carried out, write what sort of object YOu get as an answer (scalar o vector). If they out cant be carried briefly explain why not_ (To remove any doubt: are vectors; k is scalar constant, f is scalar function and F is vector function:) (u Vf div(kF) gradF curl(F)



Answers

In Problems 15-18, find a scalar a so that the vectors $\mathbf{v}$ and $\mathbf{w}$ are orthogonal. $$ \mathbf{v}=2 a \mathbf{i}+\mathbf{j}-\mathbf{k}, \quad \mathbf{w}=\mathbf{i}-\mathbf{j}+\mathbf{k} $$

So for the proof, we have to split it into two parts. The first part we have to prove is that zero effected times U equals zero and then the second part, we have to show you times zero equals two there. So if we just beginning that here, doing the first talk, we come to zero vector product with you. But you're going to say is a B. You can say that is equal to just have a company and normal dot product. So it would be equal to zero comes a zero My mum's grave, then, because we have zero times areas there. Plus, here it comes with these, they're just hoping equals area. And so that is the first part. For the second part, we know that stuff too reverse or adjusted here. So retired Hey, to be a product with their revenge and that is equal to a brookins. Was it in a row? It's the brackets because there, by the same reasoning we have something comes in Europe here. Sometimes over the answer, the whole thing equal there and that is both parts proved

We wanna do the operation on vectors U and V. So they start with you, plus B. So you pushed. He would be. I won I plus zero i from being so This would just be I j class native J. We're just being zero j or zero. Then we get zero k from you. Plus, you have to Kate, to get I find this to kay. Now we do. You find this, be we get one eye minus your eye to get I Yeah, j mind this New year, Jay to get to J, which is plus because it's positive to Jay and I get zero k minus made of two K and then salamis. I would get positive. Okay? And for three U minus half e spectacular through you first. Three times, three into the vector U. So when we were get three eye for us three j Let's do half fee. Half into victory. There. You Ah j minus. Okay, so now we do Three new minus half B is equal to three. I'm Linus, your eye, which is three i three j minus. Later, half Jay. Let's convert three into Tractor boy, which would be nine we should be six over to Shay. So six over to J minus. You have j will be plus one of the seven over to J. And then we get your okay, My loose native case where we get plus okay for our answer.

Butter V has initial point to be negative one negative three and important to be to one so is given by 34 So it's you know about her. It's very bad by the links with we were the links off we is three square has four square square roots, which is five. So Director is going by three four over five, which is three over five, four or five. This is the answer.

Okay, so of activity has a mission point negative. Two comma five and terminal points. Three. Common one. Let's find Inspector. It's equal to three plus two. Common. They get one when it's fine. Excuse me. Five Common Negative six. Now I need to find the magnitude. It's a good thing. Square root of pi squared plus six squared, which gives me the square root of 61. So my component form is be over the magnitude of B. That's it. Go to one over a square bit of 61 times five Commenting on six, which gives me five square bits of 61 over 61 comma, six square roots of 61 over 61.


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