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SB
Numerade Educator

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Problem 57 Hard Difficulty

(a) Find parametric equations for the line of intersection of the planes and (b) find the angle between the planes.

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

Answer

(A). $x=1 \quad y=-t \quad z=t$
(B). $\approx 15.8^{\circ}$

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Video Transcript

they're asking to find the parametric equations for the lines of intersection of the plains and also to find the angle between these planes. The equations of the planes are number one express wipers that equal to one and number two express to wipers to that equal to one. Now for questions A we have to find The parametric equations for the line of intersection of these two planes. So first we have to find the normal vectors of these two planes. So normal vector and one for the plain one is equal to the coefficients of each other coordinates of this plane, that is equal to 111, and similarly the Normal vector of the second plane is equal to 1- two. And in order to find the no these two vectors cross product. The values of n. one is 1, 1, 1 And the values of into is 1- two. Therefore, after calculating the cross product, pick it. Value of the normal vector is equal to the values 0 -1, 1. So in order to find a parametric equation of the lines of intersection of these two plane, we have to find a coordinate point of these two planes point of intersection. So the coordinate point of the line of intersection of these two planes can be evaluated as Yes. Okay, If you put the value of zero who said then the first equation takes the form Explains why equal to one. And the second equation takes the form Explains too why equal to one. Therefore by solving these two equations subtracting these two equations, X and one gets canceled. And therefore value of Why is equal to zero. So if we put this value of Y in the equation one we get the value of excess equal to one. Therefore we got the coordinate point for the line of intersection that is 100. Therefore in order to find out equation parametric equation. Oh the of these planes? Yeah that's right. Yeah bigot the equation as Yeah one into T into k cab. So after finding this equation we can write it equal to icap one into I cab -T into Jacob Plus T. in two K Cup. And in order to find though parliamentary equation we put the coefficient value of each of the direction vectors as X, Y and Z. Therefore the parametric equation. Off line of intersection. Yeah. Of these planes are X equal to one Why equal to minus T. And that equal to see. So this is the answer of the question. A in the given question. And next in the question they're asking to find the angle between these two plains. Mhm. So in question be they're asking to find the angle between these planes. Mhm. So in order to find the anger where to find cost heater. That is equal to the scalar value off and one vector dot and to victor who divided by and one vectors killer into into victor's killer. Mhm. Mhm. This is equal to one plus two plus two. All divided by route under one scripless Once privilege, one square hole into Route Under one sq Place to Square Place to Square. This is equal to five By three row 3. Therefore, in order to find the angle, pita is equal to cause inverse of five x 3. Route three to find out diluted in degrees. The corresponding value to this angle is equal to 15.803° and so. Mhm. Mhm. Yes, the value of the degree Rounded up to one decimal place is equal to 15.8°. So the answer to question number B is 15 point a degree. This is the angle between the two planes in the given question.