Question

Question 12 Find parametric equations for the line of intersection of the planes - 2y+z= 1 and 2x + y - 3x = -3. Does the line L intersect the plane 2x - y - 3x = 1? If so, at what point? Note: This is the review exercise at the end of Lecture 2.

Answer 1

هہ has equation of a plane (Point - Normal form) The plane centaining the point (2109 ,, 7 ) and having a normal vector ñ =(a, b, cy equation, a(2-24) tbly-ji) +(7-1)=0. gren equation of planes are, 2-2y + 3 here ñ ={1, -2, I> aid 22+y-32-42 here na =(2, 1, -3) first + 7 = 1 0 vee hare to المنع ñixña og (* -2 - 2 ů (5-1) = (-3-2) + 2 (1+4) sit sŷ +52 - - 3 Now, putting ă= 25,5,57 2 =0 in o and we get -2y+tal y-37=-3 by soking these two eque - tion ere gal, Z=1 and y = 0. .. the point point of intersection of these two planes is P(0,0,1) A line to vector for J = (0, 6, c) passing through the point (21,3,7) is green by the parametre equetiens 2.2.24 tat, y=g, tout, Z=Z, +Ct :. X(t) = 54 Y{+) =5* Z(A) = 1457 is the parametive eque of the line parallel the

öd parametic equation of the line of intersection X(t) = 5t, y(+) 25+, 2 (t) = 1 + St. plane 2nd part guren 22-y-3771 putting a=st, yost, za1t5+ we get, 215+) - 54 -3 (1+5+) - ) => 10t -st-3-15&al. >) st -15+ 24 in . ) i. >) -10z 4 ) **- :. 2(+)2 st HA) zst 22-2 27 y = -2 Z(+)2 2 litst > z=-1 the line es i. the point (-2,-2, -1). from a Now, from 2(-2)-(2)-3(-1) -4+4+3 z3+1, :.(-2,-2, -1) is not the point intersection of line and plane u

igro, line L does not intersect plane 2x-y-32=1 the