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(5) What is the length of the vector (1,1,- 1) in 9 dimensions? Find a unit vector u in the same direction as and unit vector W such that v W = 0. (6) Consider the ...

Question

(5) What is the length of the vector (1,1,- 1) in 9 dimensions? Find a unit vector u in the same direction as and unit vector W such that v W = 0. (6) Consider the vectors 81 (1,1,1) ,82 = (0,1,1) , 83 = (0,0,1)_ Determine the vector b = 381 + 4s2 + 583 _ (b) Write b as a matrix-vector multiplication Ax (A is a matrix; & is the column vector with components 31. 12, T3 (c) Compute Av using the dot products of the rows of A with & Consider the following vectors Wt (1,2,3) , W2 (4,5,6) , W3

(5) What is the length of the vector (1,1,- 1) in 9 dimensions? Find a unit vector u in the same direction as and unit vector W such that v W = 0. (6) Consider the vectors 81 (1,1,1) ,82 = (0,1,1) , 83 = (0,0,1)_ Determine the vector b = 381 + 4s2 + 583 _ (b) Write b as a matrix-vector multiplication Ax (A is a matrix; & is the column vector with components 31. 12, T3 (c) Compute Av using the dot products of the rows of A with & Consider the following vectors Wt (1,2,3) , W2 (4,5,6) , W3 (7,8,9). Find linear combination that gives %1W1 + T2W2 + T3W3 (0,0,0) with *1 =1_



Answers

Verify that the vectors $$ \mathbf{v}_{1}=\left(-\frac{3}{5}, \frac{4}{5}, 0\right), \mathbf{v}_{2}=\left(\frac{4}{5}, \frac{3}{5}, 0\right), \mathbf{v}_{3}=(0,0,1) $$ form an orthonormal basis for $R^{3}$ with respect to the Euclidean inner product, and then use Theorem $6.3 .2(b)$ to express the vector $\mathbf{u}=(1,-2,2)$ as a linear combination of $\mathbf{v}_{1}$ $\mathbf{v}_{2},$ and $\mathbf{v}_{3}$

So in the given question we have two vectors which are given us nine I. And five I plus 40. And we are told to find the dot product and the handle between these victors. So if we are given two vectors, Hey, Hey, What? I Let's be one J. And we have to find the dot product of this victor. But take with another victor given us a two I I plus B. Two J. We can find it us even times A two plus B. One times we took. So this is how we found, find did not product. Right? So we can write the first victor as 95 plus zero J. Because there is no J component. And the second victor asked 55 plus four days then that our product will be equal to nine times five plus zero times four, which would be equal to nine times five, which is 45. Right? So now that we found the door, correct next weekend. Find the angle Between these vectors using a formula which says cost it is equal to eight or be divided by the magnitude of a times magnitude of P. So the magnitude of a vector say given us you, which is given as excite lists. Y day can be found as the square root of x squared plus Y square. Right? So using the same formula, we can find the magnitude of Nyon I, which would be equal to the square root of Nyon square and there is no components. So nine square block zero square which is fine square itself. So The answer would be lion itself, right? The square root of nine square is nine. And and then we have to find the magnitude of 55 plus four J. Which will be equal to square root of five square plus four square, which is equal to the square root of 25 plus 16, Which is equal to the square root of 41. So now that we have all the values we can write costly to is equal to 45 divided by magnitude of the first terrorist nine and magnitude of the second matter is square root of 41. So when we take 45 divided by nine Times Square root of 41, We will get they answered us 0.78. So then we can write three days equal to cause inverse off 0.78. And when we take the cause inverse of 0.78, we will have the angle Theta is equal to 38.74 degrees. So this is the required angle which is in between this victors and the door product of this victor's is 45. So I hope you understood the method. Thank you.

In this question we have to practice a. Plan A and B. And we have given the sum and difference of these two that we have to find a vector A and B. In the first part. As a plus b. is 11 x minus Why? It was five years ago. When it would magnify this plus lemon by plus 90. If we are these two if we are these directors but you can find here and these plus and minus course will replace being management. We can say no he will get away Here which will be close to one of 566 less than why? Yes 14 from this dividing it but to you will get better A. Yeah After a well vehicles to three X. Last five while bless sounds this is the victory. Now putting doctor in any of these who will get up to be if I have after HSB if I put in a place maybe we can find it to be as it's all right here we're going to be close to as of today Level one. Is that the 11-? I can write it as 11 minus three x minus one minus five less. Yes -1 -5. It will be why And five G. Earned challenges the community crash five g minus 70. So we need to get to be As Yemen one is 3 paychecks. So there's a story there and grab it. Eight x minus six Y. And my nurse cruise it This will be enough to be. I've got these directors. I find it this was a part now the second part I have to find the angle between A plus B and A. For that first of all I have to find the magnitude the magnitude of A plus B. Will be equal to some of the square of all the value we can become to level sphere plus one spare spice. Their hold on the group. I will do it directly and it will be opposed to. Yeah. Wait a second. Under Rule one for December. What I will get here. It's under Rule 1 14 7. Hard to Find No Baby Food of the 2nd 1. It will be forced to again same thing. It will be able to three square plus five squared plus seven square hole in the roof. And this will be close to on the road to a teacher. The second thing I never need Is eight or B. Shepherd. Good here. Now what is A. Or B. A. Or B. Maybe close to. So it is not A or B. I have to be very clear. Maybe I should go down here. It will be mm less B within the dot. This is what I have to find here. A press B. I have 11 eggs minus right. That's five. You see issue that yesterday. It must be. And the other director is three eggs plus five. Why? You are seemingly the darker. That will be closed to that. Wanting to tree is turkey tree minus five. You guys 35. If you do the sun, You will get 63 Now we are here at the final part, right, will get us the fire answer. What we have to find is the angle between them. If I right here for us, He did not suppose. Heaters, angle At the dark, dark hair was 63. Just excuse to you or Underworld 1 47. The magnitude of a plus b. multiplied by the magnitude of aid. That was DTs. Thank you. And you can use calculator To the 63, divided by undertook 1 4% and 283. And you will get cost hater 0.57. You will get causing words and the answer will be 55 buying good. I can be that I have decided. 15 5.2. Think the disadvantage of the second part

Hello I'm your pizza for this question Let me extend this for self you know give him given component of vector paid component of victory along the X axis. It is represented by A. S Which is 4-6 and along to Y. Axis which is given nine. and component of Victor belong we have sex is is given 6 7 and component of Victor Victor be around. Why you exist is given -3. Similarly component of electricity along the X-axis is given zero and component of electricity around the why you exist it is given -6. So we have to find when you have a plus b. I mean there is a plus B A B C. And it is forwarding the question we have to solve those questions. So left rather diagram for this question. So with the help of designer we can understand what being no vector is it will do a X by plus a Y. Into the component of vector around where taxes That's competitive belong to buy access so we can write it as six that's 90. Similarly for Victory we can. Right 78 minus he's yet. Mhm. Better see do I? Yes do I minus 60 minus 60. No or we have the plane picked up a policy. What question we have to find victor. A plus victory. Yeah victor is 68 plus 90. And vector B is seven. I minus treated this seven party minus is it? No you will solve reporter. So six plus 78 that will give 39 Plus 9 -6 Yep We will see the jet of 9 -3. It will be six gee. So the magnitude of vector A plus. Sorry we have to find a component of victory. It was B. So here X. Component of vector A X. Component of vector A plus B is equal to 13. And why the component of vector A plus victor V is equal to six. No good question B. We have to find A minus closing victor A minus put into victory. C. No Victor 86 I that's mine, jay -2 into Victor. C. is 08 minus 60. No further solving this. I I didn't take the item one. So here's a zero plus 6 76 plus nine minus 60 year minus minus plus. And it will be +99 J. 19 yet and two in 20. It will be zero and minus two to minus six. That will give plus we're into. Yeah So we can write six I plus 25 to the so x. component of victor A minus bluesy is equal to six. And boy component of vector A minus pussy is 25. No we will solve it for victor C. But the bar seat we have to find victor A B minus vector C. So value of victory. Value of Victor A six I bless 90 68 plus 90 and value of victor beads. Stephen I my guest DJ minus vector C zero minus 60. So they will take items a calm place and jaden on that another place. So six plus seven plus zero 46 plus seven minus zero into fight left. Nine minus three Plus 6-8 now six plus seven. That is equal to 13. I yes nine minus three Uh plus six. So that is 20 plus 20. So the magnitude of vector A plus B. My legacy is equal to 13. It is along the x axis. And male the victim. If this victor b minus victor, C. And uncle why you access that is equal to. Well now they've been solved for be part or depart party we have to find victor. A Plus one point to victory minus three into victor. See now substituting the value of a victor, victor victor. See so here A six I last nine J plus one by two, vector V is seven. I minus D. J -3 into vectors is zero, I minus 60. No further solving this. 68 Plus seven x 2. Can do I zero into I. Okay 40 nine inclusion -3.2 into they. Yes. Well to 60. No taking I common in first time. So here in uh in this so here six plus seven by two is 3.5 into. I am plus here we will take a common Then it will be 9 -3 by two it makes 1.5. Yes 18. No further solving this six plus 3.5 that is 9.52 I Yes 9 -1.5. That is 7.5 plus 18. So 7.5 plus 18. It believed 20.25 point five in two days. So X. Component of a plus one by two B minus crazy A plus one by Toby. My next Crazy X component is equal to 9.5 and y component of victor A plus one x 2 victory. My next three and 2 victors. It will you. 25.5. That was like tradition to this question. Thank you.

In this question we are provided with three vectors their dad that that is minus of four. I and plus three J. And web lb is the way and minus of the and victor. Are these six? I minus of 70. Okay. And it is told that there is a relation between this A. B. And our victim. That is already request to care time. Sophie let's mm times. Okay so we have to find your the value of K. And some will go put all the values here. My men sit side minus seven days as it was to care time. Sophie mens minus for the last three day and less sometimes of the these do I minus day. Okay now we have to make the product of gay and I'm here so I got minus four K. I the last three K. J. Last two am I And minus M. J. Okay now I take by common so after taking I common here we get the coefficient minus four K. And for us to um and if they collect the coefficients of J then we get here today K minus M. Okay now I have to some barrier. Both the sides and after comparing both the sides, we can see that the constant coefficients must be equal means minus four K plus two. Um is coefficient to buy that must make was to six and three k minus M. He's coefficient of J. That must be it was 27 Okay, so these are my two accusations. Okay, and you can see that from equation second we can write the value of them, that is three K plus seven. And we have to put this inoculation one. Okay, after substituting in incubation one, we can see that we get minus four K. List ways of three K plus seven equals to six. Okay, after solving it we get to minus four K. Last six K. Last 14 it was 26 This provides me to case it was too minus of it means case it was too minus of four. Okay so this is our value of K regard. No use the relation that is Emily goes to three K plus seven so we should get the value of them from them means three decades minus four plus seven. Okay after solving it we get the value of them and that is I can see that minus all five. Okay so I can say that no the world will gain. This question is minus four and value of him in discussion is minus five. Okay thank you.


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