Question

Problem 3 Consider the lines ti(t) = (1,0,–2)t + (1, -3, 2) and (t) = (0,1, -1)t + (2,0,1). (a) Find their direction vectors vị and v2. (b) Are the given lines parallel? Are they orthogonal? Explain your answers. (c) Find a parametric equation for the plane spanned by vị and V2. vers (d) Find a vector that is perpendicular to both vì and v2. (e) Find a Cartesian equation of the plane containing Vị and vŻ and passing through the point P = (1, 2, 1).

Answer 1

Problem l, Lt) = (1,0,-2) + + (1,-3,2) 1218) = 10,1,-))x +(2,0,1) (a) Direction are . vectors of line l, and da e = (1, 0, 2) l's - 10,1,-1) t.

o The lines are parallel if direction rats is same rako 2e a a = b = baca and perpendicular ifa aat biba teica=0 Now here agal, bizoc1=-2 a =0 bz=1 c2 =-1 - anazt bibz tica lxotox 1-2x-1 2.to lines are not orthogonala Now. I tot -2 / - . lines are not parallel.

(62 parametre spanned equation of by Vi and a point in span { 6, 1423 so there are t r t R sol tuit rez The System System a can be written as

R37 2R, t R 3 o 1 z ton R 3 & Roz R2 2+2x+ o o | zt2x + y Now ☺ has solobon iff anty + z =0. This ħ the equation of Plane

a berpendiculon to A both To = Vector 6) t, and u vi xus = ĉ (0+2)..-ġ (-1-0)tike (1-0) b = zů tŷ tiê (e) To = (2,1,1) cartesian eariation of plane containing line le and do and pensing through poine pa (1, 2, JJ 12-1 Y-2 2-1 o - - (21-1) (-180 + 2) + (7-2) (0+1) + la 12 1-0) T = 2(x-1). + 14 = 25+ (2-1) . PHILIPS

o = 2(x+1)+(y-2) +1(2-1) 2x + y + z - 2-2-1=0 2x + y + z-s=0