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Problem 2What are the three conditions under which the following is a correct statement: (AxB) x B=Ax (B x B)...

Question

Problem 2What are the three conditions under which the following is a correct statement: (AxB) x B=Ax (B x B)

Problem 2 What are the three conditions under which the following is a correct statement: (AxB) x B=Ax (B x B)



Answers

For the following exercises, determine whether the statement is true or false. Justify the answer with a proof or a counterexample.
For vectors a and $\mathbf{b}$ and any given scalar $c,$ $c(\mathbf{a} \cdot \mathbf{b})=(c \mathbf{a}) \cdot \mathbf{b}$.

Hello, everyone. So in this current question, we have equation that is X squared plus BX plus C yes, equals too explicit in two x plus B. So we had to choose which option is correct. Okay, so let's start with expanding the light and side like what we can do in this situation is we can first expand the light inside and then we can compel the coefficients of X both sides like so let's do it. We can expand the light inside. This will become equal to access. Quell plus X plus BX bless a B, which is equals two. We can regroup it, so we get a plus B into X. Bless you, baby. And on the left hand side, we already have access squared plus B explicit. See? Okay. Now what we can do is we can compel both sides the coefficients we can compare it. So let's compare the coefficient of access square. This one here also this one the coefficient affect says he'll be and he and the coefficient of access a plus B. So music was to a plus B. Right now the coefficient is see here and here The coefficient is a B like so C is equal to a B, right? So the finance solution is b equals two a plus B and see Sequels to maybe which is matching with didn't see right, so the correct option is see?

All right. So you have We have to. Victor's be cross brought everything. He's still. Is it true that you see some real number? Is that equal to is what? The plane, the scene, but a and then in the cross product. That would be so, uh, is that equal? Well, let's see. So we have a hey, peoples, too. So a one put it, excluding 82. Uh, a three. But is he coordinate and B goes for Well, we'll be one. Be one too, You three. What is in that product? Well, that probably is gonna be ableto he's competing to determine that within Matrix. I j Okay, so these, uh, coordinate representation. Is this same my saying before the eye complaint he want, So we wanted them. So I plus a two times J on that, plus a three downs. Okay. Deserve those air degrees forming ended Victor's I j k for the coordinates. Ex wife on see eggs. Course. Right. Why the day guards point to see escape. So, um, they did cross part is gonna be the same. Was computing determinant of these matrix one two. If we Then it was real. Here. One two three so that, well, the girl's part up here is gonna be They're turning on this. I dance these times that mine Is that that? So it's high times Hey, do with them be three, two. The three. Mom. My ears. Yeah, a three hands, too. Well, uh, for the shape of wanted, he's gonna be ableto Jimmy. So it's the zone. So you know that it's a one. It really minus you want to? So J Times one the three. My news A three. You what now for the cake. Oh, boy, It's a call to that's the blue. We're good king. So you remove that doubt? She had a want EMS. You do minus B Want them to. So that is a one. Two miners. Two. And be one. No. Well, what happens if I will deploy I or that stuff, but I see Well, well, it only gets multiplied by she's It's gonna be a sea here. C m c in there. See? So these part. But he's, uh they were fun. Side this equation known. So what happens if we do? They're right. We did the right hand side of the question. So you do the course product off C A Crosby as you see, or c A. She will be going to with the plane. See on each component. You can see one. See Time's too. See Time's Really So why did you do that? Cross product off. See, a bean is gonna be doing the determinant off I Q. And then see, Time's a one. I see them too. I don't see them say three, you know, with one two. Good, Good. So what did properties Italy determinants Seems it seems that the term not these linear image let's roll his role on that grow grow. So that means that we can be there is circumstantial to play in the hole, bro. We can pull that girl stomped out it's gonna be able to do to turn you not off. See, Time's for removing the constant out of the road. So as you can see, there will would have these, uh, two. This is precisely that. Because so here is the same because you can't see decide Didn't rent, right? Onside. This agree? You know they're equal. Where's you been? Left on site. So So? Yeah, the very go on there You can prove that by using the properties of the determinant this linear, I wonder what defecation you know. Back, constant buddy. A real number. See riel Lumber back multiplication over the whole road. So by the

So we can start this problem by using the foil method on this, um, kind of condemned on the condensed, simplified version of the original equation Explicit a X place to be so when we do used to form, that's what we do at times actually gives just x squared when we do the inside. I mean, the outside fo outside we do plus b x, the inside hi. Begin at post a ups and then l the last two times we have cost a B Then one little thing that we can do here, that we're really kind of show If they answer here, we can take these two middle terms here, and normally we can combine them. But because we have A and B instead of normal number coefficients what we can do instead, we can keep that X squared. Nothing we could do with that. But for those two military, we can go ahead and factor out that acts and what we have left is B plus A. So all we have all we did there. What set up our equation here at the bottom, where we have some number, whatever the some of B plus dais times APS just like we have appear. So the two generalizations we can take from this if these two things are equally we know they're equal because all we did was expand or condensed equation are condensed. Um, expression are here into that the least. If the equation is true, up top in the equation has to be true for when we after we expand. So they still think we can make two observations here. First thing we can look at the third term for each of our two expressions. So we have a B a Times B equal to see there the third term. They have to be equal. We have a A Times B is equal to Capital City. We can also make the assumption based on the second term, that be pushed A or what we multiply by X is equal to what we multiply bags up here, which is Capital B. So we have a plus. B is equal to Capital B, and we look at our options part of their that just still happen to be option C. The attempts BC go to Capitol see an A plus B equals Capital B. So just based of the two assumptions that we can make. We can tell that the right answer to this problem if you let her.

For a fact aeration off the expression X squared plus B explosive is equal to X plus a multiplied with experts Be if this is true and this can be true only ven if see is the multiplication off A and B and capital B must be the summation off a plus B s o out of the four options. Given the right option, he's option, see?


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