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Problem 58 Hard Difficulty

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

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


(A). $x=1+5 t \quad y=1+11 t \quad z=7 t$
(B). $85.9^{\circ}$

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Video Transcript

in the question they are asking for two. Type of question. Question number is find the parametric equations of line of intersection of the plains and question B is find the angle between these planes. The equations of these planes are number 13 x minus to a plus and equal to one and number two to express by ministries that equal to three. In order to find out answer to the first question it is asking too find the parametric equations of the line of intersection of these planes. So solution to question is first, we have to find the normal vectors of these two planes, so no victor to play, money is equal to 3 -2, 1, and normal vector of the second point is equal to 2, -3. So in order to find the equation, we have to first find out normal to the plane That is equal to N one cross and two victor. That is equal to the cross product formula. According to the table. The value of anyone is three minus 21 And in two is 2, -3. So the value of the normal vector N is equal to five 11 7. And after this we have to find the coordinates at the point of intersection. Therefore coordinates at the point of intersection Of these two planes. Yeah, can be found as follows, if we put the value of Z is equal to zero. In equation one and two we get She X -2 is equal to one and two weeks less. Why equal to three? So in order to compute the value of X and why? We have to equate these two equations. So we have to multiply the second equation between order to cancel out the term. So this is after multiplying with do we get so the after adding these two equations, why terms get cancelled out and therefore seven X equal to seven? And the value from here For X is equal to one. And yeah, if you put the value of X equal to one from above then value of Y is equal to mhm by solid from any other two equations, we get the value of Y Which is equal to one. Therefore we found the coordinates the at the point of intersection of these two planes to be yeah is equal to yeah 11 zero. Therefore, in order to find the parliamentary equation at the point of intersection of these two planes. Yeah, yeah. Is equal to the coordinate less. The normal vector component for the corresponding direction vector. So one plus 5 d I cap Plus one plus 11 T. Jacob plus zero plus 70 K cups. This is equal to in more general form to find a parametric equation. We can say that the corresponding coefficient of the three direction vectors I J and K caps are the values of X by nz. So this is X. This is why. And this is it. And hence we find a parametric equation as X equal to one plus 5 50 Y equal to one plus. Eleventy. And there is equal to 17. This is the answer to the question A. Of the given question. And the next question they are asking to find the angle between these two planes. So the formula to find the angle is caused citizen were to the scalar per that of the values of normal vectors. And divided by the the scalar product of the individual normal vectors. Right? So the value of course tita is equal to Yeah From the values of N one and 2 as evaluated before it is equal to and one is equal to mhm three minus two, one and into Vector is equal to two, one minus three. So you see catholic top dot product. We get Capacity equal to 6 -2 -3 divided by route under nine plus four plus one multiplied to nine route and the nine plus four plus one. This is equal to One divided by 14. And therefore Peter is equal to cause inverse of one x 14. Mhm. So the value of the pita in degree is equal to 85 point nine 044 Degree. The question they're asking to find out the value of theta rounded to one decimal place. So this is equal to 85.9°. Therefore answer to the second question is Tita, is he going to 85.9° and this is the answer for the angle between the two planes