two-vectors-a-and-mathrmb-have-the-same-magnitude-m-but-vector-a-points-north-whereas-vector-b-point

Question

Two vectors, A and $\mathrm{B}$ , have the same magnitude, $m,$ but vector A points north whereas vector B points east. What is the sum, A $+\mathrm{B} ?$
(A) $m,$ northeast
(B) $m \sqrt{2},$ northeast
(C) $m \sqrt{2}$ , northeast
(D) $2 \mathrm{m},$ northwest

Sachin Rao · Accepted Answer

this problem. We have given two vectors A and P have the same magnitude. M but Victor eight points north. This is Victor A and Victor be point east. This is activity and we have given magnitude of Vector A Is a cult group magnitude of vector B. What is the some A plus v. So this will be cool to vector a vector. We visit Kyoto and the road Victor, a square less vector V square. So we know that magnitude is same, which is M square. So this will be called to the route to into him. And if we talk about the direction, direction, this angle will be 45 degree and we can say this is M route to so vector sum is equal to m route to north east. This is the answer. Oh, thank you.

Ze-Han Lee · Answer

So, uh, in this problem, we have two vectors, uh, on a, uh, the vector A points north. So this is a vector A and victor be points east, and they have the same magnitude. So that combined combined vector will be pointing to this direction. So this is an all faced and a lance off this equal square root. Uh, time Specter, eh? This is the magnitude off the combined of actor. Yeah, So we can see that the solution is, uh ah, b. Okay.

Zachary Warner · Answer

part of our questions says we have two vectors, vector and Vector B and were asked to use unit vector techniques to show that the magnitude of their vector sum is given by the square root of a squared plus B squared plus 2 80 co sign thought. Okay, So for part A, we&#x27;re going to say that the results in Vector Victor are is just the sum of the two vectors vector a plus vector B. So then the dot product of our with itself. So we&#x27;ll do our daughter our is going to be equal to hey not be or excuse me. A plus b vectors dot a plus the vectors well, are are, of course, is just gonna be r squared fix this year. Okay, so are not ours are squared. And that&#x27;s because the dot product of two vectors is there magnitudes multiplied by each other. Time of the coastline of the angle in between. But the coastline of the angle in between two of the same vectors is the coastline of zero. So it&#x27;s just one. And so it&#x27;s just there to magnitude multiplied together. So its magnitude of our times, the magnitude of our which is R squared. So the dot product of any two of the same vectors with itself is just the square of that vectors magnitude. And now we have a dot Be a plus b dot a plus b. So this gives you Aye, aye, aye. So we just have to go component by component. So vector radar vector a plus vector a dot vector B for us vector b dot vector A plus vector v dot vector v But as we just mentioned, the dot product of two of the same vectors is just the magnitude of those two vectors squared. So eight a is just a magnitude of any squared be dot B is just magnitude of b squared. And then what is a dot Be a dot b is a times b times the coastline of the angle in between five but be dot a is B times eight times the coastline of that. So those were the same thing. So you end up with this two times a a times b times the co sign of the England between which we&#x27;re gonna call five so are squared is equal to that Therefore are is equal to the square root of that. So are is equal to the square root of a squared plus B squared plus two time&#x27;s A claims be comes the co sign. If I next in the square root to include all of this and that is asked what we were asked to show for the solution to our question in part A so we can go ahead and box. I did as our answer. Okay, now part B start a new page here says, uh, if A and B have the same magnitude for which value of five where the world will their vector son had the same magnitude as a or B. So we&#x27;re told that a M B had the same magnitude, so vector a magnitude will ignore the vectors. So magnitude today is equal to the magnitude of B is equal to the magnitude of the result in Vector arc. So I&#x27;m going to replace everything in that previous expression we found with a right because everything is or we could replace it with our Doesn&#x27;t matter. We could replace it with the they&#x27;re all the same. So but we have is a square is equal to, so I replaced r squared with a squared, Um and we had, if you go back previously we had our squared is equal to a squared plus b squared. So this is equals a squared plus b squared. But we&#x27;re gonna place be squared with a squared because they&#x27;re all the same plus two times a times B. But we&#x27;re gonna replace me with a again. So is two times eight times a Texaco sign a Fine. So, um, one of these a squared on this side is gonna cancel with one of the ace words on that side. And then we can subtract this a squared over to this side, right? And then divide by this a squared here. So what we end up with is the co sign. If I is equal to negative 1/2 because the ace where it&#x27;s cancelled, we can take the inverse co sign of both sides to sell for five. We find that five is equal to 120 degrees. We could box, and it is their solution for party

Zachary Warner · Answer

our question says that I had is in the East direction, which I write is E. J Hat is in the north direction, which I write is in, and K Hat is in the up direction, which I just write us up. And then I also wrote this little wheel here. I like to use this wheel because it&#x27;s a nice visual representation and simplifies the solution to cross products of perpendicular vectors. Since these air all perpendicular to one another. No, I&#x27;ll just draw this little three dimensional diagram here. So you see what I&#x27;m saying? So this is K hat. It&#x27;s up. Makes a right angle with I hat east Makes a right angle with J hack or north, right? So all use red here. If you go in the cock wise direction, it gives you positive. If you go in the counter clockwise direction, it gives you negative. So I cross j is K. But Jay Cross I&#x27;m moving in the opposite direction is minus K. So you can just see k cross I then moving in a positive direction gives you j. And then, uh, I cross kay moving in the negative direction. Give you minus J Okay, so real simple. All right, so let&#x27;s go ahead and use that, and we will begin. Uh, part, eh, Let&#x27;s do it on this page. So we have a little bit of room on the bottom. Let&#x27;s go back to black here part, eh? And never wanted to know everything below that line And on the other pages, as solutions to the question. So this is J hat cross minus. I had right, Because the question says find North crossed west. Well, north is J Hat west is the opposite of east, so that&#x27;s minus. I have. Okay, well, you can pull the minus sign out front. So we just have negative J cross I j cross eyes minus k. So minus minus cake. Is kay hat just plus cape work a hat. And that is a but their solution for a Okay. Now, for B, be wants the dot product of minus K hack dot minus J hat because it wants the dot product of downward, which is the opposite of upward and south, which is the opposite of north. Okay, so pull the minus signs out front. They canceled. So we have the dot product of que hot with Jay happ But the dot product of any two perpendicular vectors is zero since Kei hair is perpendicular TJ hat. Okay. All right. Now, for part C were asked to find the cross product of I had with Kay hat or, uh, east with up. Well, I cross kay is minus j moving in the opposite direction on that wheel. Right. Well, plus J is east, so minus J hat is west. Uh, excuse me. Plus J is north, so minus J hat is south. Okay. Part D wants us to find the dot product of minus. I had with minus I hat, since it wants us to find the dot product of West with West and I had is east. Well, the two minus signs get pulled out front and minus times minus is plus to the minus. Sign cancel. So we really had the dot product of my hat. I had Well, the dot product of any vector with this self is one. So this is one. And then lastly for part e grass to find the cross product of west with west or excuse me, um, south with south or minus J hat cross minus J hat again minus science canceled. So you have Jay had cross J hat, those, uh, cross product of anything with itself a zero. So this is just zero.

Sri Datta Vikas Buchemmavari · Answer

the given Wickers, our A and B is vertically downwards and is eight units long on B. It&#x27;s fifteen years long on makes I mangled thirty degrees. Is there not? Which is. And if I do access now, if you choose our exact sis to be horizontal on by excessive vertical as usual, we can see that waiter okay would be minus eight, Jake, because it is completely vertical but is in downward direction, which would be minus y tension and find us a check on people. Be because it makes an angry sixty degrees with X axis. We can write it. Ask fifteen cost sixty. Excuse me. I can&#x27;t plus fifteen Signed. Sixty Jacob. We&#x27;re playing this into calculator. You can see that it is seven point five less thirteen jigga now for party. We need to find a less. Now we know that when we are workers, we are their confidence, their individual components separately. So here, because it is just minus A j, eh? Just minus eighty. We connected us place, Cyril. I can&#x27;t. It is. It&#x27;s x component is single. Hence, by adding their two ex competence, we can spread it as zero. Let&#x27;s seven point five. I can&#x27;t less minus eight. Let&#x27;s thirteen Jacob, because these two are divine confidence off Andy Now zero plus seven point five gives a seven point five less five chacha. This is our table Now we can find out its magnitude is invite a grocery in it will be the square rolled off the sums of the squares of the individual confidence and that will be five square. Let&#x27;s seven point five square. I never squeal. Horn route, it stands out, Toby, If you wouldn t know after later, almost nine years now the angle it makes with the X axis would be turning worse off the white. Confident we should be five divided by the competent. Just seven point five. It&#x27;s turns out to be thirty three one seven degrees. This will make sense because if you see that victor A plus B seven point five five let&#x27;s Fight J. We were expected to look something like this. Hence this angle would be about thirty three degrees. Now let&#x27;s go toe r v I&#x27;LL be is finding director B plus, eh? Now we do the same thing again. We are individual conference, you know the bees ex compliment is seven point five. Andi is ex con. Print is still I don&#x27;t want to fight with a nightcap. Now we are BJ confidence. You know that be confident is thirteen on ex Con J. Conference is minus eight. It might be like a chicken of this again. Turns out to be one point five. That&#x27;s five. Jake. We know that Vector Edition is coming. Video on. We&#x27;ve showed it short this here now as usual because this vector is the same as the last victor. It&#x27;s linked will be the same as the last one, which would be the square root off seven point five square. Let&#x27;s fight square, which would be nine. And the angle it makes it except sis Botanic. Worse off. Ex confident. I&#x27;m sorry. Why? Confident? Divided by the ex Confident it would be terribly wanted. Degrees This victor would also look something like this just the same as the last victor. Now for party, we need to find a minus B. Now we do the same thing again. We subtract the individual components this time so we know that the ex competent off a cereal and if I confident of B is seven point five. So we subtract that from zero on. Jake confident off It was minus eight and the Jakob print off B was hurting. So it subtract that. Forget it. Minus oven. Want footage? I have Les. I&#x27;m sorry. Minus twenty one. Cheek up. Now we know that this vector lies in the fourth quarter because both of its confidence are native. No, we can find the angle. It makes it exactly Stina, which will be done in worse off the confident do whatever ex con print, which is twenty one by seven point five. It stands on Toby, seventeen point four degrees. On the length off this vector would be the square root of the sum of the squares off the individual confidence. It should be seven point five square. Last one. He wants fair. It stands out. Toby. Twenty two point three units. No, the food part is finding a b minus. See? So we do the same thing as before, But this time you subtract confidence of a from conference. You know that he&#x27;s ex component is seven point five on is in conference is zero. I can&#x27;t now we know that these ex conference is thirteen on is ex conference is minus eight. So minus my two minus has become less honest. Answer, Toby seven point five. I have less twenty one jigger. You might know that this is equal to minus. They want to speak, which we will expect you to scale her multiplication ofthe actors. Now, this victor would have the same magnitude has a previous victor because the individuals in the science of the individual components don&#x27;t matter. When you square them, the magnitude of the spectre will again be some point five square. That&#x27;s twenty one squared square hole our half, which is the whole road on that one again be twenty two on three degrees. I&#x27;m sorry. Twenty two one three years by this vector instead of being in the third quadrant would be in the first quarter. I&#x27;m sorry. I said the previous victor was in the director a minus B was in the fourth quarter, but this is actually the tar quartered. This is the fourth quadrant. No going back this victor because it has positive X and Y conference would like the first quadrant on the angle it makes with XX is and again be fine. Ono does tannin worse off. White confident doing debate X confident, which are twenty one divided by seven point five, which again turns out to be so unrefined for agrees, that&#x27;s it.

Mitchell Cutler · Answer

berate walking, docked. So this problem you service goes that hopefully you&#x27;ve been understanding and developing over this over it. These videos over your time during physics and math. All right, so in this problem, you have three. Vector is the 1st 1 helical vector thing starts at the origin groups or something like that from in England. 35 degrees with the X axis. It has the manager of 145 units. Um, firstly, the units off, they&#x27;re all right. Second, victory forms in England. 65 degrees with the critical with a vertical line. Great. Was north the third vector starting from that point. First inning glove, 15 degrees with the vertical. All right, so in this problem, it here we want to solve for the magnitude of this is Victor a right here, Victor, be Victor. See, you want to solve for the the magnitudes of Victor being Victor. See? So, um oh, please. He was triangle right here. Kind of buried in a lot of lines and markings, but we&#x27;re gonna do a lot of things with this triangle right here. It&#x27;s a pretty, pretty special triangle, I guess. All right, So we&#x27;ve got, um, your first. We need to find each of these English right here. This one, this one and this one. So we know that, um, this single rate here, just this one part of the angle, it&#x27;s going to be 1/4 15 degrees because this single right here&#x27;s 15 degrees and their along parallel plant, uh, lines with an intersecting line so the inside angles are gonna be equal to each other. So this right here, this little part of the gang was 15 degrees. Uh, this part right here, we confined by doing 90 minutes 35 which is 55. Which means that that fall angle, they&#x27;re merely drawing trying over here with her measurements. Who&#x27;s drive the train going? I&#x27;m gonna draw on the board too. Well, it&#x27;ll be a challenge, but I have faith that it&#x27;ll work. Marriage. So this thing right here was 55 plus 15. That&#x27;s gonna be 70 degrees. All right, this single rich here, we can find that by taking over here, we have the straight lines about 180 degrees minus, um, 65 degrees when this thing we&#x27;re looking for. But it&#x27;s this angle groups, ministers angle right in here, which we can find that angle fixing that Lansing. That&#x27;s when you degrees 19 my just 35 rather 180 minus 90 minus 35. That gives us 55 degrees for this Ingrid here. If we do 180 minus 65 minus 55 that gives us 60 degrees. Then once again, we have another triangle ridge right here, which has a total of 180 degrees inside of it. So if we do a 180 minus 60 minus 70 we will find that this single ridges went to be equal to 50 degrees. All right, And we have this magnitude right here, 1 45 units. So using the lob signs, we confined, the magnitude could be in C Whoa. It was a strange murky right there. In case you were running that was supposed to be in a all right, it&#x27;s a magnitude of eight is 145 spending the managers of E and C. We can use a lot of science than that is we have a side, they said right here over the sign of its angle, which is the angle opposite of that side. So this I right here, missing right here. It&#x27;s gonna be a court to the that same idea aside, over an angle anywhere else on our triangle. It&#x27;s a pretty nifty law. If you didn&#x27;t know it, now you do. We&#x27;ll find be first we could find. See? First, it doesn&#x27;t really make much of a difference. Um, miss choosing it to choose it. So we have to be in the 70 degree angle causes the That&#x27;s the side angle combination that are opposite each other, promoted by both sides by sign of 70 implied that end your calculator will find that be is equal to when ingen 77.87 units. You can do the same thing for side. See, So I know the angle that&#x27;s you&#x27;re looking anything was 60 degrees right here and again, much playing both sides by sign of 60. Then plugging in inter calculator will find that C is equal to 163.92 units. All right, and there should circle those right for blocks around it. There are your answers

Problem

If $\mathbf{F}_{1}=-20 \hat{\mathbf{\jmath}}, \ma…

Problem 1 Easy Difficulty

Two vectors, A and $\mathrm{B}$ , have the same magnitude, $m,$ but vector A points north whereas vector B points east. What is the sum, A $+\mathrm{B} ?$
(A) $m,$ northeast
(B) $m \sqrt{2},$ northeast
(C) $m \sqrt{2}$ , northeast
(D) $2 \mathrm{m},$ northwest

Answer

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