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Find the rectangular coordinates of the points whose cylindrical coordinates are (a) $(4, \pi / 3,2),(\mathrm{b})(5, \pi,-4)$

Calculus 3

Chapter 29

Partial Derivatives and Double Integrals

Section 2

Curves and Surfaces in Three Dimensions

Partial Derivatives

Campbell University

Harvey Mudd College

University of Michigan - Ann Arbor

University of Nottingham

Lectures

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05:15

Plot the point whose polar…

02:23

Graph each point in a pola…

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Okay, so this farm wants you toe convert a couple of Solyndra coordinates into rectangular coordinates. So a solid record ascender. Thus, Linda cornice are in terms off our data and see where Arteta is. Ah, the polar coordinates of the projection of a point onto the X Y plane and Z is just ah directed distance from the X Y plane to the point so we can use these formulas here to convert the cylindrical coordinates into into rectangular coordinates. And you can notice these two for mills. These two foreigners him used convert polar coordinates which are in two dimensional into McCain record minutes. So we have here for pie thirds negative too. If we were to plot this, it would have in our value of four. So it would be a distance of four from the pole. So 1234 beard distance of four from the pole. The data is pie thers, which mean it rotates. The point is rotated pi thirds radiance from the polar axis, which is the positive X axis. It's rotated counterclockwise. So if we were to rotate pi third, that would be somewhere around here somewhere around here and this would be Piper, is this. There's some angle. And finally Z is negative two, which is the distance on the Z axis. So negative to would be two points downwards. So your point would be somewhere here in that quadrant. So if you were to convert this into you can either coordinates, we need to use the form owes. Let's do that To calculate X, we have to use our coastline data where ours for and data is pi over three. In this Sequels to to cause co sign a pi over three is 1/2 before time. One has to exactly why I need our sign up that also ours for data is pi over three. Sign of pi over three is Sarath 3/2. So you would end up with a Y of tour at three. And finally disease Josie souls you would equal to negative two. So our answer in terms of X y Z would be to come a two round three comma negative too. This would be your final answer for part A. If you want to check on the plot that we moved a distance of two on to move this insult to on the X axis and we moved a distance to on the Y axis a distance of two. On the the negative tone of the access, you will find yourself in the same region of as we plotted. So that's how you can check to make sure we did everything correctly. So this is part B. Part B is converting the courts under coordinates to negative, however, to one into retaining their coordinates. For his part, this first So our is ah, value of two. So I mean it is a distance to from the poll data is negative pi over tomb. That means we're rotating it clockwise instead of counterclockwise pirate to radiance or 90 degrees so it would be rotating to so it would rotate to the negative y axis. So that would be an angle pi over two. And finally, it's ah, the value of one has the value one. So that's the value one. So your point would be somewhere here above battling. So now that we've gone in our point, let's convert it in to retain their coordinates. So X equals to our coast. I take it off where our is to and co sign off. Beta. Martha is prior to goes on. A pirate too is zero. So we have an X value of zero. Why? Why is equal to To sign off data where there is negative O negative, Piper Till so sorry I messed up here. That should be a negative Colson. A coastline of negative high over to but thats zero so off sign of negative. However, to is negative once or two times a day. And one is negative. Two for your wife value. Finally, these just seem so Z equals to one. So your answer in terms of X Y Z is zero you too one. And if you look on the quiet, you can see if we it is indeed zero on the X axis. Make it two on the Y axis and then one on the easy access. So that's how you can tell me Did that correctly

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