Question

Let a, b, c, d and e be the first five digits of your La Trobe student number. i.e. if your student number is 12345678 then a = 1,b=2,c=3, d = 4 and e = 5. Use your appropriate values in the question below. 2 a Consider the line. L = {(2,3, 2): ?- } y-b 3 2 (a) Write L in parametric form. (b)0 Find the coordinates of the point P at which L intersects the plane z = d (recalling that d is the fourth digit of your student number). (1) Find the equational form of the plane Si that is perpendicular to L and which contains P. (iii) Let S2 = {(2,y,z) : x + 3y + 2z = e}. (where e is the fifth digit of your student number). Does S intersect Sy? Justify your answer. (iv) Find the distance between S, and S2 (C) Find the equation of the plane S3 which contains L and which has the form fx + Oy+gz = h, where you must find values for the real numbers f, gand h. Hint: There is a very simple way to find this equation using the equational form of L.

Answer 1

Ane Civen that 2-3 2 zt let) 3 ye atat 2= 3+at, x-12 y 2 e 4-3 times L {1,8,291 2-a obis where a=1, 6-2, 6-3, day, e-5 " then equation of line al 98- Li x-1 x-1 4-2 then parametric for will be x= 1+t ( civen line is passed through point Me (a, b, c) that is passes through 11,2,3) also it cuts the pat Plane at z=d that is 224, putting into equation treted T- az lt Ž 2 32 - Get y=2+ 3 % so the points where it cut the plane I (312) (ii) Equation of plane passing through 14,8 p) amel perpendinetay to line yba zre zme is given by. I lala-) tho lyryp) 4 n/2-2220 so equation of plane possing through P= 1 3 1 1 2 14 and perpendicular to line List 3 T ara n

cata 1(2-2) + 33- Z)+ & 12-4)=0 x+3y +22= 20 This to the equation of plane si passing through a and perpendicular to L. = = (1) Given de 210,9, 2) | & + 3y tazze si then equation of plane e par x+3y +2225 Angle between plane e, and so 1.1 +3.3+2.2 12+3225 179722 Cos oz u oz cost W o or at " These two planes are parallel to each other. in They does not intersect each other. ( we knows distance between two planes. ayx+by+G2 td, 20 and axt by+G2t do to is di-da Jae+2²+G4 so distance between s. and so 205 J 12 +3²+22 so distance between s, and so is 15 Tu Is 15

(vi) It plane ext my teed contains the line x-24 y-y, 2-2, then. a 5 lat mbt ne=0 and bout my, +nzed (0) have to find equation of plane fx toy+gz=h BO daf, mao, nag and dich. we 3 then plane passing through L: *! io. from equation (9) f. 1+3.0+ 9.2=0 ftag=0 and folto.atg. 3=h ehf +39=h det g=th then fa-ph and tas equaction of plane. &xt dy-z-8 -24x toy hazh &x=2& -ax+2=1 which is required equation of plane where f=-2, 9=1, hal