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Point) Express the point given in Cartesian coordinates in spherical coordinates (p, 0, $) . Note: you really only have to do the work for one, if you use a little ...

Question

Point) Express the point given in Cartesian coordinates in spherical coordinates (p, 0, $) . Note: you really only have to do the work for one, if you use a little geometry and your knowledge of the trig functionsA) (2 V6, 2 V6, 2) =BJ(- ! V6, % V6, 2) =C(? Vo,_9 V6,_2) =DJ(_ 9V6, _ 9v6,

point) Express the point given in Cartesian coordinates in spherical coordinates (p, 0, $) . Note: you really only have to do the work for one, if you use a little geometry and your knowledge of the trig functions A) (2 V6, 2 V6, 2) = BJ(- ! V6, % V6, 2) = C(? Vo,_9 V6,_2) = DJ(_ 9V6, _ 9v6,



Answers

Find the Cartesian coordinates of the following points (given in polar coordinates).
\begin{equation}\begin{array}{ll}{\text { a. }(\sqrt{2}, \pi / 4)} & {\text { b. }(1,0)} \\ {\text { c. }(0, \pi / 2)} & {\text { d. }(-\sqrt{2}, \pi / 4)} \\ {\text { e. }(-3,5 \pi / 6)} & {\text { f. }\left(5, \tan ^{-1}(4 / 3)\right)} \\ {\text { g. }(-1,7 \pi)} & {\text { h. }(2 \sqrt{3}, 2 \pi / 3)}\end{array}\end{equation}

Okay, ladies and gentlemen, So in this problem were given a whole bunch of options. I mean, a whole bunch of different polar cornet pairs, and we're basically find the Cartesian coordinates of each of the points. Um, thing is, this is relatively a trivial process, but let's let's let's do them one by one. Okay, let's do them one by one. Part a is route to PIRA for where What we're gonna do is we're gonna first to draw it out. So we know that are the radius is route two, and we're at Piper Force that this pi over four. If this is route two and is the x axis the Y axis, then as you can see the x axis, I mean, the x corner of each of this point is just, you know, route to co sign if pi over four, which is to what's cause on the part before I think it's rude to over to. Okay, so this is one. What about the y? Coordinate? Well, why? Cornyn is gonna be rude to sign this pi over four, and this is also gonna give you one. Okay, so we're done with a be 10 Remember, you might want to draw the point here. You think this is 10 and you would be correct? Because this is the 100.10 Is this point because the radio distances one and the angular distance is zero. So we're not moving away from the X axis. So this is 10 and there's the X axis. The Y axis. The ex cornet in this point is in this problem is very trivial defined. It's the X corn, and it's just one. What about the white corner? Well, as you can see in the picture, the white going to zero. Okay, let's be Let's do see, we'll see. Is zero in pirate too? Well, zero and poverty to remember. The first number gives you the radio separation away from the origin and in this case, is zero. You know what that means? That means that you're not moving away from the origin, your at the origin and when you're at the origin, it doesn't really matter what angle you are because you're the origin. So your at this point it does not matter what angle you're because you're just the point. So the X corner zero and same as the white corner. Okay, what about Deep? De is negative route to impart before? Okay, this negative route to negative route to when remember counterclockwise is positive. This is negative pie before I mean Piper for So I will be here. Negative route to end pi over for Okay. The X coordinate of this point is you just do your coasts. I write you just do your It's the X cornice negative route to, um co sign it pi over four, which is going to give you negative root two times to over two, which is negative. Two over. To which a negative one. That this is negative. One x Cornish negative one. What about the y? Coordinate? What will the white corn it is just gonna give you negative route to sign of pi over four, which is negative. Route to route to over two, which is negative one. Okay, so this looks like the same thing as a but but negative this x this. Why? OK, we're done with the less duty he is the negative three and five pi over six. This is negative. Three five pi over six is like, really closed. Two pi to 65 or six, which is pie. So it's really close to this point. But, like, not quite. We're pi over six. Short this and goes five or six. Okay, so it is negative three. And this is also so okay, this point is the negative three and five pi over six. Because where he started all from here. And we rotated five pi over six. This is five Pi over six. This this point is where we ended up at We're gonna find the X cornet while the X cornett is That's first remind ourselves that this is three this distance. Um, the X coordinate is three co sign of pi over six, and that's he could have three times. Let's see rat three over to Okay, this is three rat 3/2. Why corn? It is three Sinus pi over sex, which is three times want of it through return. Okay, this ex corn and this is the Weichel. Okay, this is a This is E. What about F? Well, f is five are tan. Let me write it this way, Tad inverse for over three. Who this one looks a little bit spicier than before it was five. While the angle is tangent universe for over three. Does that mean that means this angle is later? You know, tanginess tangent. What is hand intending? This sign of her co signed. That's Ah, I was it over adjacent, right. This is the opposite. And this is the adjacent this is for and this is three. Apparently, I'm not drawing it correctly. This looks like it's for or over three. Okay? And and there's distance is five. So this angle is gonna be tangent universe of for over three, right? Yeah. Okay. Just I just did a little bit of a mental check and see, that's correct, because that is that is correct. Um, we So Yeah. You know, the most important thing is to have this picture, because as soon as we have this picture weaken just trivially read off the X corner because X cornices three in the white corner. This for All right. Okay. What about Jean and negative One and seven pi. This is negative. 10 Nearly 17 Pi is the same thing as negative one. What's seven Paice of empire? It looks like it's that looks like it's pi plus to buy time. Three. Right? I can ignore this Two pi times to me because you know, too 500 series is just did your multiple of to buy. It's not gonna change anything. Um, so this is the same thing. It's negative One pie. At this point, this distance is one. Okay, so access one, why is zero okay? Because the way I did this is because, you know, I started all from here. Negative 10 There's a negative 10 angle angle. A zero r is negative one. And I don't want to move seven pi because that would just take me forever. I just I simplified the seven pi into pie and two ply plus three. I mean to two pi times three because six pi plus by seven pi and I ignored this six pie. Because wherever you are in the graph, if you had an intradermal tipple of two pie, you're gonna end up by the same point so you can always ignore it. Always ignore it when you see it in the promotable to by being added to something because that doesn't do anything. This doesn't do anything. So I am left with something that actually does something, that something that actually has a new effect and that, in fact, it's more rotating by pie. Okay, All right. I am at this point now, and I rotate it by pi, which is 100 degrees. I'm gonna be at this point. So that's how I did it. Okay, um so as soon as you have this picture, you could just read off the X and Y axis X and Y coordinates. Because exported it just one. And the white corn is zero. This is the answer for part. What? What is this, G? Yeah. What about H? Well, h is to root three and two pi over three. Okay, um, the strong is out. Two pi over three. This is this piper three. There's a two pi over three, and this is 23 and two by over three. The X. Okay, this is this is pi over three. This is another pi over three. There's another partner. Three, this is This is the singles pi over six. Because thes two angles have to be the same and two tons high over 32 times. Pirate threes. Pirate. I mean, two times five or six is Piper three. Okay. Um so this angles Piper six. The egg cornet of that is just gonna be to root three. Sign of pi over sex, which is to root three 1/2 which is Route three. Why of that? Has to do. Three co signed of poverty six, which is to root three. Let's see Route 3/2, which is three. We're done. We're done for this problem. We're done from a all the way to each. Thank you very much. Has seen the next video.

So we are given the spherical coordinates of a point and rest to plot this point. Then we're asked to find the rectangular coordinates of the point. Spiritual coordinates of the first point are too sure. Hi over to so pi over two years now to graph this. Yeah. No we're gonna draw our X, Y and Z axes. Alright kid. Right be sending now because data and fire are both between zero and pi. Well sorry fi is pi over two. So we're in the xy plane. In Theta is pie or two. So we're in the first quadrant of the xy plane. Yes. Yeah alcohol sweet. It's a bit like oh B C. Well that's you know that's Evan. What? Right. A lot of fun. Well James was the since data equals pi over two and five equals pi over two were on the positive Y axis. So the point is located about here. That is the. Now find the rectangular coordinates because they do this graphically. Or we could use the formula for spherical coordinates. So we know that X equals road times the sine of phi times the co sign of data boxing. It's a sex. This is two times the sine of pi over two times the co sign of pi over two, which is zero. Why is equal to rho times the sine of phi times the sine of data, which is two times the sine of pi over two Times The sine of pi over two, Which is equal to two and Z is equal to rho times the coast Sine of phi, which is two times the coast. Sine of pi over two which is zero. Therefore in rectangular coordinates the point X. Y. Z. Is 0- zero. Mhm. Then in part B. Were given The point in spherical coordinates for negative pi over four. Hi over three. Never fucked kids. It's just getting one asked with the this island in the caribbean right? Like. So we have that row is for data is negative pi over four and five is pi over three. Since five pi over three where above the Xy plane. Since data is negative pi before this means we're above the Xy plane and in particular were above the fourth quadrant. The words that that Since Roe is four, we're at a distance for from the origin. They were like Olivia guns. Old boyfriend faked his own death. Mexico, yep. Yeah. Come on Bruce. Yeah. Mhm. Well, nick you hear about this? I'm doing research. Read jesus christ. Great. We've been taught our engineer. Yeah. I don't know what faked his own That he drowned like 10 years next year. Just the Yeah, avoid like child support. She was. Yeah. Louise. Yes. Finally yes to say yes. So first of all, rotate hi over three on the positive Z axis, negative pira for princes here, however, three of them we go up to about ah to hear maybe and then we're going to be a distance of four from the origin. What? Maybe about here? Like first of all, what I love too is so in green. This is the position of our point. Yeah. His point is cadence. You're got it. Now. To find the rectangular coordinates. Will use this spherical coordinate equations. We have X equals row times The sign of five times the co sign of data. This is four times the sine of pi over three times the co sign of Negative Pi over four. This is four times no Route 3/2 times Route 2/2. This is Route six. Why is equal to rho times the sine of phi times the sign of theater, which is four times the sine of pi over three times the sine of Negative Pi over four. This is four times right Route 3/2 times Negative route to over two, which is negative Route six. Finally, Z is equal to rho times the co sign of fi, which is four times the co sign of pi over three, just four times one half, which is to therefore in rectangular coordinates. The point is that X. Y. Z equals Route six, Negative Route six two small, and those are no the most.

Problem is a lot of the point. Who's all the coordinates are given? Then find there's a Cartesian coordinates off the point before, eh? Our is to Andi. Their eyes three power too. Three pie a word too. Yes. Year Is this three pile right, too. On dh Laugh our asses able to to this point this point as to three pi over chill. And now, as a condition card, Innis axe is a cultural our terms We'LL sign Ada So call sign three power too is Cyril. This is Cyril. Why is you go to our Tam's sign Data inside three. Power to his neck to one. This is negative two. The Kardashian card anus. Is there o negative too? B Nina, Is pi a war or this is pi over four. Ah, nde are is beautiful. No, this is a point is yeah. This is a tough too high over war. Now axe is cultural. Are, um school sign? Hey, they're syncing contributed to but of two over to success is one. Why is you go to you'd have to sign. Our war is his also. What is a Kardashian card in it is one one One thing a fate I cz negative. Opie over six. Was he? This point is one negative high over six. So, Nick to Juan. Hi. Over thinks. Yes. This point. This pond is negative. One next to high over six now, actually is you go to our terms. It was signed. Negative. High. Over. Thanks. Next. One times you have three over to This is negative. Fruits of three over too. Why? It's illegal to make two one times signe negative high over six. This is the cultural negative. One times makes you half justice. Half she is a Kardashian card in it is negative. Roots of three over too and one half.

It's already converting from spherical two rectangular. So it's important to note that there is a conversion for this, we're gonna have X equals row, which right is p sign fee code saying there you go. And then why is going to equal rho times sign for the signed data and then z is going to equal roco saying feet. So we're going to use these equations to answer all of our other equations with the given points. So for example, if we're given 45 or six power before there is just an example but we can see this with any problem that's given to us. We would end up having x equals four times the sign of number four which is route 2/2. I'm the code sign pi over six Which is going to be route three over to. That's going to end up giving us route six as we sit here. And then why is going to be four Times The sine of fee? Again, that's going to be rude to over two and then we're going to have times sine of data which is going to be one half. So really that's just gonna end up being rude to because we have the fourth, so this one has so Can you cancel and that's just going to give us route two And lastly we'll have Z which is going to equal Uh four times the co sign of the so that's for times route you over to and that ends up giving us to route to as our final answer


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