use-the-accompanying-figure-to-write-each-vector-listed-in-exercises-7-and-8-as-a-linear-combination

Question

Use the accompanying figure to write each vector listed in Exercises 7 and 8 as a linear combination of $\mathbf{u}$ and $\mathbf{v} .$ Is every vector in $\mathbb{R}^{2}$ a linear combination of $\mathbf{u}$ and $\mathbf{v} ?$
Vectors $\mathbf{a}, \mathbf{b}, \mathbf{c},$ and $\mathbf{d}$

Kush Khamesra · Accepted Answer

All right. So to do this problem, we want Teoh go to point a through D by moving with Well, the only the vectors v and you from the origin. This deals with the parallelogram rule for addition and is only possible because u and V are both linearly independent. So by this logic, what that means is a would equal a eagles You minus two V. So we&#x27;re moving you vector up and to the left to V be with you. Cool. Two u minus to V. We move to you vector up and then to be to the left. See? Would you go to you minus 3.5 v And last but not least, d we&#x27;re equal. Three U minus four v again, These are all just moving from the origin from 00 The origin to these points using only the vectors you envied moved to this to this location of point A, B, C and D

Runpeng Li · Answer

Okay, so for probably eight, we need to observe the the bigger in our next move. Eyes. I&#x27;m not people to allow the entire figure on this white board, but how to try to explain how, uh how do how do we sum problem? So first we need to find out. How do we defuse W in terms of U. N. V. So every review we observed the program involved in W, then we will find to be and some of to be a you, we&#x27;ll be vast action. And for that tracks and again, we need to find out the andi be corresponding her low crime. So this will be to you and for us to be, because the negative to you and the to be will be the happens off a corresponding parallelogram and why it&#x27;s over. Why, it will be, um, half off the so we first still have next to you. But, um, for V, we can take two in the two plus half. So that&#x27;s 3/2 be. So that&#x27;s why Oh, wait. I&#x27;m sorry. Three should be three plus one house. That is 7/2. Yes, I&#x27;m unhappy. Okay. And for her vectors, I&#x27;m dizzy. So the course woman in parallelogram. I observed the graft. It will be for a fee. Yeah, for a fee. Plus three of you. So it&#x27;s minus three. Open. Thank you. So here&#x27;s how are you?

Bridget May · Answer

Okay, So today I&#x27;m gonna show you guys how to use the dot product to solve this type of problem. So the problem says usually given vectors to find v times W and v times re. So your your first thing you want to look at is what is the dot product. So if we define the dot product, the DOT product says that if you have a times would be that&#x27;s gonna be equal to a one. Be one plus a two be too. So you might be wondering, what exactly is a one? What&#x27;s B one? What&#x27;s a two would be, too on. So, you know, if you have even more numbers than that, it&#x27;s gonna be greater. You can have a one B one plus a to B to pull a three b three. It all depends on the vectors you&#x27;re given. And when we look at these two vectors here V and W, we have, we can see that we have only two numbers for each of them that we&#x27;re gonna be working with. So to define what a one and be one is gonna be, let&#x27;s put it in terms of V and W. So if we put it n v and w terms, we&#x27;re gonna look at a dot product that looks like this be times w equals V one, w one plus V two w two. So if we wanted to find what the woman w won our first, we&#x27;re gonna look, let&#x27;s look at V and Viner v terms. So this seven right here is gonna be the V one because it&#x27;s the first number that comes with a variable. Our second variable is J. Again. We have only two variables, which is why we define the equation as V one w one plus three to W two. If we had three, three variables say we had plus to Z, then we would have two more. But for this we only have two. So we&#x27;re gonna have V one and V two. When we look at the W, it&#x27;s the same thing. So negative three is our first value. So this is gonna be W one and it says negative J. And we&#x27;re gonna pretend there&#x27;s a one there cause we know that it&#x27;s negative one times j and that&#x27;s going to get us w two. So if we put this in we do the terms. W. And we defined V one at seven, which we&#x27;re gonna put there, and w one we said was negative three. So we&#x27;re gonna plug in negative three, and then V two, we said was negative, too, because it&#x27;s really important to not forget that negative sign. So negative, too. Times W two, Which is gonna be negative one. So if we multiply this out, we&#x27;re gonna get V W. He&#x27;s cool. It&#x27;s a negative. 21 plus negative two times negative one. We know when we have two negatives multiplying, it cancels out. So we get a positive and two times one is two. So negative 21 plus two is equal to negative 19. So we&#x27;re halfway done solving this equation. So first, we&#x27;re gonna go look at the second part we have to solve for which is V times be so we&#x27;re gonna define it in the same terms. So we&#x27;re going to say the times V, just like the dot product is equal to V one, and V is the same thing in Savi one plus V two v two. So if we plug this in, we&#x27;re going to get the times V to equal seven times seven plus negative to again. Don&#x27;t forget that negative times negative too. So if we simplify this, we are gonna see that the Times three equals seven times seven. So that&#x27;s 49 plus negative two times negative to the the negatives. Cancel out and you get plus four. So 49 plus four is equal to 50 three.

Jason Li · Answer

Hello, given vectors. U seven negative teary in vectors V 16 real on. I performed these for operations on them and we want to graft a result. So first, let&#x27;s start with a Hey, we have is you. So us seven Negative too. Plus B, which is +16 and doing this component wise we get this is equal to eight and maybe, uh, eight and four l for B. We have again seven negative to for you. Minus 16 Again, we subtract component wise. So disc is US seven minus one, which is six and negative U minus six, which is negative. Foreseen. We have a scaler multiple, So we&#x27;ll do that. So two times. So then negative too. And we have plus 1.5 one point, uh, one six and now doing scaler. Multiplication for us. This gives us 14 negative for all complained to in show us 1.5 nine, multiplying three over to win. And now this gives us adding 14 plus 1.5, which is 15.5 and negative for plus nine, which is I&#x27;ve now for teeth. We have Southern negative too. Minus two times 16 again scaling application first. This gives us Southern negative, too minus 2 12 Now we just subtract and this gives us nothing but five and negative. 14. Not a graph. Everything. Let&#x27;s do a first. So 84 we move, hate to the right and then for up. And then we&#x27;ll draw a line from the criminal. Where the origin to the terminal point now for B six. Negative eight. Move six to the right for the X coordinate and negative eight down for the white quarter Nets over here again. Watch us trial a straight line, but effective on a narrow See, we have 15.5 and five. So 15.5 to the right, a little off the chart. But that&#x27;s a and in five. So around here, if we extend good five Well, look, something like this. Now 30 we have the X being five So five to the right and the white being negative. 14. So $14. So over this will be higher terminal point in yellow. So we&#x27;ll just draw a line all the way from the origin to the terminal point again, giving us

Yuou Sun · Answer

and we want to rewrite this factor into in a combination off I n j. Right. So minor Simon, which is the horizontal couple? It is just the question if I so just minus seven I and also minus one j So just ignore this one here. We can just you know it. So there should be minus seven. I minus J.

Himani Sood · Answer

Hello, everyone. This is in the money. And Shea, we&#x27;re gonna talk about having subtracting and multiplying vectors. Okay, So, um, in our problem were given to Victor&#x27;s you and be Let&#x27;s go ahead and write down both of those. So you is given as, um, 73 and of the is given us negative. 73 please. All right, so for part a, it us is toe add you and be Let&#x27;s go ahead and use red for this. Okay? So when we add vectors, we basically just add the X components together and the white components together. So, um, seven plus negative. 70 in three plus three is six. So this is our answer to part A, which is you. Plus to be ok, part B, us, us to subtract U minus E, which is basically the same thing, is in part, a been stuff adding recently, subtract so seven minus negative. Seven is seven plus seven, which is 14 and then three minus 30 So this is our answer to Part Beach. Okay. Part C asks us to do to you plus 1.5 e to you. Most 1.5 would be okay So when we have like to you, for example, all we do is multiply ish. Um, each component by the number radical efficient. So to you would simply be seven times two is 14 and three temps to a six. And so, um, like, in the same way 1.5 b would be, let&#x27;s see what? Seven tons, 1.5 iss, 10.5 and three times 1.5 is 4.5. Okay, Okay, so now that we have to you on 1.5 b, all we have to do is add them together in the same way that we did Part a, um And so for that over here to do is your 14 plus 10.5, which gives us 24.5 comma, six lists. 4.5 is 10.5. And that&#x27;s our answer. A man for part D or we&#x27;re going to dio. Okay, so the question is humans to be and so we have to calculate to be on the side, and that would be seven negative. Seven times two is negative. 14 and three times to a six. And then we just have to subtract u minus to be so seven minus negative. 14 is seven plus 14 which is 21 3 months. Six is negative. Three. So that is our answer for parts ABC. And I hope that was helpful. And I hope you have a great day.

Problem

Use the accompanying figure to write each vector …

Problem 7 Easy Difficulty

Use the accompanying figure to write each vector listed in Exercises 7 and 8 as a linear combination of $\mathbf{u}$ and $\mathbf{v} .$ Is every vector in $\mathbb{R}^{2}$ a linear combination of $\mathbf{u}$ and $\mathbf{v} ?$
Vectors $\mathbf{a}, \mathbf{b}, \mathbf{c},$ and $\mathbf{d}$

Answer

Each vector is a linear comhination of $\mathbf { u } + \mathbf { v }$ since:
$a = \mathbf { u } - 2 \mathbf { v }$
$b = 2 \mathbf { u } - 2 \mathbf { v }$
$c = 2 \mathbf { u } - 3.5 \mathbf { v }$
$d = 3 \mathbf { u } - 4 \mathbf { v }$

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David C. Lay, Steven R. Lay, Judi J. McDonald

Linear Algebra and Its Applications

Grace H.

Catherine R.

Anna Marie V.

Kayleah T.

Absolute Value - Example 1

Absolute Value - Example 2

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Each vector is a linear comhination of $\mathbf { u } + \mathbf { v }$ since:$a = \mathbf { u } - 2 \mathbf { v }$$b = 2 \mathbf { u } - 2 \mathbf { v }$$c = 2 \mathbf { u } - 3.5 \mathbf { v }$$d = 3 \mathbf { u } - 4 \mathbf { v }$

Linear Algebra and Its Applications

Grace H.

Alayna H.

Caleb E.

Kristen K.

Absolute Value - Example 1

Absolute Value - Example 2

David C. Lay, Steven R. Lay, Judi J. McDonaldLinear Algebra and Its Applications

Grace H.

Catherine R.

Anna Marie V.

Kayleah T.

Each vector is a linear comhination of $\mathbf { u } + \mathbf { v }$ since:
$a = \mathbf { u } - 2 \mathbf { v }$
$b = 2 \mathbf { u } - 2 \mathbf { v }$
$c = 2 \mathbf { u } - 3.5 \mathbf { v }$
$d = 3 \mathbf { u } - 4 \mathbf { v }$

David C. Lay, Steven R. Lay, Judi J. McDonald

Linear Algebra and Its Applications