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5 nd thetangle between thettwo planes [2 Points) 42+3 + 53 Jandk HNHB H2ZHEBCOSP'(#)Sin"'(#) Elo '(#)Cos (ICOr...

Question

5 nd thetangle between thettwo planes [2 Points) 42+3 + 53 Jandk HNHB H2ZHEBCOSP'(#)Sin"'(#) Elo '(#)Cos (ICOr

5 nd thetangle between thettwo planes [2 Points) 42+3 + 53 Jandk HNHB H2ZHEB COS P'(#) Sin "'(#) Elo '(#) Cos ( ICOr



Answers

Use a calculator to find the acute angles between the planes in Exer-
cises $49-52$ to the nearest hundredth of a radian.
$$2 x+2 y+2 z=3, \quad 2 x-2 y-z=5$$

Fight go same -26. Cosby equal to five, question number one then goes A Plus 13 goes big Equal to 11, equation number two solving to into occasion. Number two plus equation number one we have 25 course A -26 Cosby equal to five Plus 20 cause a plus 26 goals be equal to 22. Does this get canceled? and 45 calls say equal to 27 then cozy equal to 27 x 45. That is three x 5. If courses three x 5 then goes be equal to 11 -10. Cosette and into three x 5 divided by 13. This is to that is five x 13. So called says three x 5 and Cosby's five x 13. No Yeah A right angle triangles. okay 453 From the signing equal to four x 5 and tiny will too four by three And from this figure mhm. Mhm Yeah. The 13 12 5 Sign be equal to 12 x 13. Can be equal to 12 by five. Mhm. First option a sign of A plus B. We know that sign of a plus bees sign E. Cosby plus courses sign B. And substituting the value we get that is equal to 56 by 65. Simply put the values of sinus because be like that. Okay, no function be science square of a minus B by two. That is equal to one minus course A minus B by two. Well you know that is equal to cosy minus bees. We know coz a minus B is equal to cozy Cosby plus sign is sign B. That is 1 63 x 65. So So value is uh sign of esquire to buy 65. Okay to buy 65 now see okay, no fe minus B equal to you know that Danny minus can be divided by one plus 10. It can be mhm. That is equal to four by three minus 12 by five, divided by one plus four by three. Indu 12 by five. That is 20 minus 36 by 15, Divided by 15 plus 48 x 15. 15. 15. Get canceled. That is equal to -16 x 63. Now be six square of a -7 by two equal to to buy one plus course of A minus B. We know because of A minus B, that is equal to two by one plus 63 by 65 that is equal to 65 by 64 1 simplification. So A implies S. B implies P. See implies Q. And the in place are

Hello students. In this question we have to determine the angle. Theta between the planes that is two weeks minus Y plus two equals +23 and six. X minus two. Y plus three years equals +25 Okay so this angle theta that is costly to this will be equal to the dot multiplication of them. Okay, so to manipulate the basics, that is 12 plus uh minus one more player. But this minus two, this will be plus two and two more player by this tree. That is six Deaver Bay. Under root of this two square plus minus one is square plus this two square. Okay. And under root of the this 60 square plus minus two squared plus three square. So from here we get after solving it we will get that 20 day verb. This is equals to nine. So that is three and this is equal to 49. So that is seven. Okay, so from here we get cost. It equals to 20 by 21 so tita equals two. Course in words for our post 2021. Okay, So from the given options, option C become the correct answer for this problem. Okay, so this is the angle theta between the two planes. Okay? Thank you.

Okay. Here was solving a triggered a metric equation, which is signed five data minus sign three data. Is he coulda Coast time, fourth data. So we're gonna need a some of the product identity here. That identity will be sign a minus. Side B is equal to two coastline A plus B over to time. Sign a minus bi for over to send a for us five. Data B is three data, so we'll substitute this stuff in, so we're gonna get to co sign by Saito plus three data over to their sign. The five date of minus three data over to Mrs All he could to co sign fourth Ada. So we'll get to times co sign. We work in that. Princey. That's eight. Data on top, divided by two. So coastline full data. We'll have signed five date of minus three theaters to data divided by two. It's just saying data he could have co sign Fourth data Could bring all this over on the same side of the equals. Dropping some of the currencies here is we don't need them. Okay? We'll take out the greatest common factor, which is a coastline. Fourth data, we got to sign Date of minus one. We can set these equal to zero separately. So co sign 4th 8 R equals zero and science data equals 1/2. Okay, what is co sign equals zero. We know that's at the top in the bottom of our unit circles down here and appear so those are ah, uh, board into integer multiples of pi over two. So forth data Should he call pi over two. Let's do by times K and fourth data should he called three pi over two. Let's any into your multiple of two pi you can divide everything by full get pi over eight. Let's pi over Four times came and then three pi over eight plus pira before times K So those were the 1st 2 parts of our general solution. Go ahead and sold. This science data equals 1/2 welcome places where the y coordinate is 1/2 looks takia and here it's that they very close pi over six That's any insurgent multiple of two fire and they dare equals five pi over six Any indigent multiple of two pi. So those are the other two pots of our general solutions. So everything underlined in green here will constitute the general solution to this problem

We're looking in this problem for the acute angle between the plane to expose to why plus two z equals three in the plane to X minus two. Why minus c equals five. So sort of has a quick picture. We have got plane one here and we have some other plane plane too. Here. This is the angle between the planes. This is the thing that you want to calculate. Um, in order to do that, we need vectors and vectors that we know from these planes are the normals. So this is the normal two plain one. And this would be the normal airplane too. And if you notice this is 90 degrees So this angle is 90 minus data. This is also 90 degrees. This is 90 minus data. This is also fatal. So the angle between these two planes is the angle between their normals. So normal vector for this first plane is the vector to 22 and the normal to the second plane Is the Vector two negative too negative. One taken from the coefficients of the two planes. The dot product of these two vectors is equal to the magnitude of Vector one That's the magnitude of vector to times the coast sine of the angle between them. And so to calculate this angle, it will be the art co sign of the dot product, uh, in one and in two, divided by the magnitude of then one and the magnitude of end, too. So this would be two times two plus two times negative, too, plus two times negative one divided by the square root of two squared plus two squared plus two squared times the square root of two squared plus negative two squared plus negative one squared that gives us and, of course, the Archos on that. Sorry. So we're looking for the park co sign of four minus four minus two over the square root of 12 times the square root off nine. That is the our co sign of negative two over, uh, six square roots of three. The art co sign of six square roots. And three, uh, gives us an angle of one point 76 radiance, and we know that high over, too is approximately 1.57 radiance. And so this angle is not acute. So in terms of the picture way, have got some angle greater than running in a room, but we have some numb, some angle greater than 90. The angle between them will be this one. That would be pie minus 1.76 which is 1.38 radiance.


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