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(6 points) If u and are vectors in R" with Ilull = Ilvll; prove that u + and u - orthogonal vectorsare...

Question

(6 points) If u and are vectors in R" with Ilull = Ilvll; prove that u + and u - orthogonal vectorsare

(6 points) If u and are vectors in R" with Ilull = Ilvll; prove that u + and u - orthogonal vectors are



Answers

Suppose a vector $\mathbf{y}$ is orthogonal to vectors $\mathbf{u}$ and $\mathbf{v} .$ Show
that $\mathbf{y}$ is orthogonal to the vector $\mathbf{u}+\mathbf{v} .$

So the question we have the vector you So that has, uh, two components. Just partisan political. Ah, and question. We have been asked that projects and oh, and we have another activities have been asked protection off you along the vector B and U minus projection or expected along the huh. Or overnight. So you approve this property. So that is assumed there. This competent is you one and this confident issue just their human. And you go up to each other. No, you has to bump owners wondering Google you want is predicts enough. You were long direction on your door is projection. You in particular that accent? So we have to prove this. Let us know that. So we know the former position off you in the garrison. Obvious given us you door. But phone magnitude of these breath They may be this vector issue one and you're too u minus. This woman you talk on like need to be stressed. We have to find that you want and you toe proponents agree to that. So they're dark part of you. One dot You do you is coming out of these, you know? Then you one is order of Anarchy You open now. We could check this out So you won dark You do Is a dog work that we have before form So I could do that. But I did us you dark Be one The people of peace craft for me Dork U minus you No view A bomb maybe two of the square The performing this whole person give the next expressing us So this is This is a scale of 1 30 This person is There's a scaler quantity on here. This person is also Skela quantity So work me to do we'll take the scaler under the hope so is integrated like this you'll be then this magnet therapy Hold stress and performed the ah daughter person off you and you so you don't get be endorse you minus We have destroyed or product So we have the scaler quantity You know more Um I will be scrapped. This dumb has been market very place. So it becomes square. Kendall, we have baby doc. Okay, Now everything is Skela here because corporate gives us color quantity so now can easily control there You'll Darby and we don't read or you can return as your therapy. This will become you dork boots, Grab upon mag nickel of the square minus you would order, Be full scratch upon magnitude of to the power for and go Very dark week and better than us Magnetar of the square. So this expression becomes you dark be Oh, great! Upon Mordovia Square minus you, dog be. It's great for making a jewel of bees for this time becomes zero since dark. Product off you and you want you zero. So therefore you wanted you to a proper need groping each other. Those protection off you in the direction of the end. You're minus projects in a declaration of Rhea or don't agree to that we have okay.

Hello there. So for days exercise we got these two vectors U. And V. We're going to define the inner product with respect to the matrix A. So it is basically multiplying each of the vectors with the matrix A. And then take the usual the product that we know. So in this case the Matrix A. is the finest 2111. And we need to show that you is a phenomenal to be you remember the two vectors are functional in some space within their products. If and only if the inner product of those two vectors is equal to zero. So this is what we need to show. So let's start by computing the multiplication of the matrix by the factors in this case is 2111 time is the vector 33 And this is just the vector 9, 6. And likewise for we we have 21 11 Times a vector five -8. And this is just the venture two minus three. So this is what we need. And then we just need to calculate the inner product of these two factors. That's going to be just the product of 9, 6 With 2 -3. And the result of this is 18 minus 18, Which is clearly zero. Therefore you and we are, you know.

Way Have you last the in do u minus the equals zero. Therefore it is you, don't you? Fine. As you don't mean Plus the door u minus. We don't me was goto this after this part? Therefore we have you don't you might be a way. What is that? This is the answer to the problem.

So if you and you posie and humans you're orthogonal That means you plus v don't you mind? His V is going to be equal to zero. So let's see what happens here. So we got you got you minus you. Don't be plus v dot You migh ve dot b is equal is here. So this is really just the magnitude ofyou squared, minus you dot ve plus we can switch these orders. It doesn't really matter. Minus nineteen of the squared. These guys will cancel out get you nine to you squared is equal magnitude of e squared. That means banks interview past vehicle to the magnitude of the so they have the same, like


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