5

Herencker 17 - 20,ind olliviIlu+vllena17.U = (2.1), W=( 2)18 0 = (-3.2. 2}, V (1 4)19. 0 = (1.2 , W=(-3. 6)20 U = (2 -3.6), V = {10. 15.30821 Underwhat conditions ...

Question

Herencker 17 - 20,ind olliviIlu+vllena17.U = (2.1), W=( 2)18 0 = (-3.2. 2}, V (1 4)19. 0 = (1.2 , W=(-3. 6)20 U = (2 -3.6), V = {10. 15.30821 Underwhat conditions is Uoil+Uvllzio+In Exercises 22 # 25, find the unit vector U in the direc22

herencker 17 - 20,ind olliviIlu+vllena 17.U = (2.1), W=( 2) 18 0 = (-3.2. 2}, V (1 4) 19. 0 = (1.2 , W=(-3. 6) 20 U = (2 -3.6), V = {10. 15.308 21 Underwhat conditions is Uoil+Uvllzio+ In Exercises 22 # 25, find the unit vector U in the direc 22



Answers

In Exercises $17-22,$ express each vector in the form $\mathbf{v}=v_{1} \mathbf{i}+$
$v_{2} \mathbf{j}+v_{3} \mathbf{k} .$
$5 \mathbf{u}-\mathbf{v}$ if $\mathbf{u}=\langle 1,1,-1\rangle$ and $\mathbf{v}=\langle 2,0,3\rangle$

Hello and welcome. We're looking at chapter 12 Section two. Problem 22 were given two vectors. U and V. It wants us to find negative to you. Plus three V. And it wants it in Standard unit Vector form. That's what this is called Has the standard unit vectors. I j k. Just a reminder. I is 100 this unit vector J 010 and then kay 001 Those were three standard unit vectors s. So you could solve this problem by finding negative to you plus three V and component form and then converting into standard unit vector form. What I'm gonna do is I'm gonna convert them right at the beginning because I think it actually makes the algebra a little bit better. Eso converting you into standard unit vector for my just look at the three components X component the V one, As this notation puts it, uh gets multiplied by isis would be negative one Hi in the V to the second component gets multiplied by J plus zero j on. Then the third component gets multiplied by K. You could think of as X y Z or first component second component simplifying that negative one. I easy to write. That is negative. I zero times anything is zero. So we could leave that out, then. Just boss to Kay. So already, this is a little bit easier to write than the component form. For the second vector. This is going to be one eye plus one j less one. Okay, so all three components 1st 2nd and third are all one. So I just plugged them in to my standard unit vector form. Uh, so I could write that a little bit simpler just by leaving out the ones and I have I plus J plus K. So now I can use these. I'll start a new page. I can use these and it's actually gonna make finding negative to you. Plus three V, at least in my opinion, a little bit easier. We want to find negative to you. Plus three V where you is Negative. I plus two k and the is I plus J was K so I could just plug in. I can substitute the standard unit vector form of you and for you in my expression here on Same for Via can plug that standard unit vector form into my expression. And that's exactly what I'll do. Make sure you substitute in France sees because you're multiplying Negative to buy all of you. Not just the first term. So this is you. I just replaced you with this here. I'm gonna replace V with this here. So the nice thing about Standard Unit Vector form is I can now treat this like a new algebra problem I'm going to distribute. This is actually distribution. And then I just combined, like terms. So it makes the algebra of the scaler. Multiplication and vector addition makes it a little bit simpler for us. So if I distribute and get to I minus for K plus three, I am distributing the three all the way through Vector V Plus three J plus three K. Now, at this stage, I've got, like, terms here. I've got my to i and three I and then I have my negative for Kay and free case. So I want to combine my life terms too. I plus three eyes five I put off, keep him in alphabetic order. So I'll do my three j next thes air taking care of this is taken care of, and I just have negative four K plus three kids. Seven. Okay. Uh, sorry. Testing that seven K negative. Four plus three should be negative. Close one negative. Four plus three is negative. One s o. This is negative to you. Plus three V in standard unit Vector form on a reminder. You could have left it in component form done negative to you plus three V all in component form. And then just translated your answer. I wouldn't be any more. Work may be a little more, but I think one of the strengths of Standard Unit Vector form is it makes these linear combinations these combinations of vectors a little bit easier to calculate. So we were asked to find in Standard Unit Vector for Mega to you. Plus three V. And that's exactly what we found here.

Be given to factors in question 17 as you and be over here and we need to perform indicated operation and then we need to add the two actors. So what's the what's the way to Arzu plus week? You let's write of actors first you is negative four and three. And uh we is to a negative five. So if you want to add of actors, we have to add the corresponding parts as and we have to add the X coordinate with the X coordinate. And we have to add the Y coordinate with the Y coordinate. That's all the addition of the back to work. So if we have to earn that's going to be negative four plus three and three plus minus five minus sign will eventually come because of because it's already a minus, that's gonna be negative four plus two which is negative two and three minus five which is again negative. So this would be the required factor. You plus fee after the addition of the individual factors. Thank you.

Rest to finally possible. Two possible vectors for V. V one v two Given the following information. If we know that you dot V is equal to 10 we immediately know that to be one plus three V two is equal to 10. Solving that for V one, we are gonna end up with five minus 3/2 v two. There's a value for V one and knowing that the magnitude of the squared equal 17 we immediately know that V one squared plus the two squared equals 17. We can substitute the value we found for V one into this equation. So we end up with five minus three over to the to the quantity squared. Plus, the two squared equals 17. And when we distribute all of that, we end up with 13 V two squared minus 60 v two plus 32 equals zero and the possible solutions for V two. Then it's either four war and over 13 and now we can plug that back into either one of the other two. Either this equation here or this equation here it's up to you and you end up If that's your values for V two you're possible values for V. The 1st 1 is going to be at the Vector 53/13 comma a over 13 and the other possibility for the vector is negative. One comma four and we've solved for the two possible vectors lead.

So for a problem. 18. We know that we are going to you want going to be solving for vector Z and were given this equation You plus V plus Z is equal to zero so we can go ahead and start by solving this equation for vector Z. So we know that to do that, we're going to subtract you vector you and Victor V from each side of our equation and that will give us a vector. Z is equal to negative you minus feet and we are told from, uh, the problem that you is equal to negative 13132 Envy is equal to one negative too negative too. So we could go and substitute these into your equation. Z is equal to a negative view. So that means that we are going to be in multiplying our vector. You buy the scaler negative one in minus vector V. So again, we're going to be multiplying by this killer on negative one. And when we go ahead and do this, we can change our numbers of our specters of the components of our vectors into, uh, the essentially just negative version of whatever they are. No or positive. They started out negative. So we will multiply each term. Each component of our vector you by negative one. And we get negative one. Oh, excuse me. Positive one negative three. Negative too. And we're going to add this to the negative components, the vector V multiplied by scaler of negative one s. So that will give us negative one positive to positive too. And we know that when we are going to add two vectors together, essentially, we are just going to add each corresponding component to each other. So that means we'll add the ex components together, which will give us one clause negative one. We can add the why components together. So we get negative three plus two. And finally we will add ours e components together, which gives us negative two plus two and only simplify this. We find that Vector Z is equal to zero negative one zero


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