Problem

Determine whether the planes are parallel, perpen…

03:21

Problem 54 Medium Difficulty

Determine whether the planes are parallel, perpendicular, or neither. If neither, find the angle between them. (Round to one decimal place.)

$ x - y + 3z = 1 $ , $ 3x + y - z = 2 $

Answer

Angle between the two planes is $95.2^{\circ}$

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Video Transcript

Hello. So the question is taken from victor's and geometry of the space. And uh we uh the ocean is determined whether the planes are parallel and perpendicular or neither. If neither find the angle between them. So given the planes are x minus y plus, freezer is equal to one and the second is three X plus y-. That is equal to two. Okay. We need to check whether these two are popular or perpendicular. So in order that both are perpendicular, the coefficient of this exercise is that is a express B Y plus. Is that is equal today? That abc must be same in both cases. Or we can say that issue is same in both cases. So let me check even over to the Evan is the coefficient of first plane. And it was the coefficient of second plane. Okay, is equal to even know what they do even basically the occasion was busy all weekend. See this the coefficient of um can we say like this? Yes, we won over B two is equal to see one of us. You too. If this condition is satisfied then the plane, South Pegler and if this condition is not satisfied then the planes are not regular. And and the perpendicular condition is if the adult product of these vectors, one dot a two plus B one daughter, B two plus seven dot c two is equal to zero, then the planes are perpendicular. This is why the condition that direction consigns of both of these two planes are perpendicular to each other. Uh basically the angle between them is uh 90 degrees. So cause of 90 0 in that case the door product of these two Pegler grain direction cause I N. Is equal to zero. Okay, we are checking this boat condition for first condition. Let me check one over a two is three. First condition and this is second incarnation and form first condition one by three -1 x one. Let it be Which is not equal to went further. It is not equal to three x -1. Okay, so this condition is not satisfied. So planes uh note parallel. Now we need to check whether these are perpendicular. So in order to check that even the potato is one door three minus 1.1. B one or B two is and seven dot c two is minus 3.1 which is not three equal to three minus one minus three which is not equal to zero. So the planes are also not perpendicular perpendicular. Okay, so now we need to evaluate the angle between them. So the angle between them is the coefficient of this. Uh basically the magnitude of this ever and I plus B one G Plus C1 day does A to I Plus B two G. No plus C two. He divided by square root of a one squared plus B. One square plus c one square. In two squared of a two square Plus B two Squared. Let's see two square. Okay, so the value of this coast data will be Evans one in 2, three minus Because I daughter is one but the dot product of perpendicular vectors are equal to zero. In the similar way. B one or B two basically that is minus one and minus three. Do I do by squared A one B ones even? It is 1 -1 and three one minus one square is one square. And uh plus three square is. And four second cases +31 minus one. So that will be three square plus one squared plus one. Which gives the value three minus 30 minus one over square with nine plus 2 11, 9 plus 2 11. So minus 1/11. So tita is equal to cause In world -1/11, which is the required angle between these two plains, which I'll need about when the color and no Tyler Or let me check that value of anger which is schools in the world -1 where 11 is of course in the world 1 -1 by 11. That is 95.2° which is required. And some of discussion of this close without in. Thank you

Urvashi A.

IIT(BHU), Varanasi

Topics

Vectors

Calculus: Early Transcendentals

Chapter 12

Vectors and the Geometry of Space

Section 5

Equations of Lines and Planes

Problem