Consider the line given by :I: = 9 + 0.666667% 1: y = 9 + t and the plane with z =9t equation 7r : ~$ ~ 2y+ z = 25 (a) What is the geometric relation between the line and the plane? Answer: intersect in point 3 (b) If they are parallel at a distance, calculate the distance. If they intersect in a point, find the point of intersection. If the plane contains the line, enter 0. Answer: C] (Remember to enter a point using parentheses, such 35(1.2,3)J (1 point) The plane tr is defined by the vectorparametric equaen 7r : :'(s,t) = (5, 5, 7) + 3 (0,4, 5) + t (4, U, 5). Find an equation for 'II' in general form. Plane equation: C] = 3+ 2t Given the lines l1 : y =t -7 and Z = -3t = 2t - 9 12 : y =t - 9 Z = 5t - 9 (a) find equations (in general form) for the parallel planes 71 containing 1 and 72 containing 12 71 1: 71 2 (b) determine the geometric relation between the lines: they are skew CGiven the skew lines 11 : y t and z =5+t :1: = 17+ 5t 12 : y = 17 2t 2 = 20 + 3t (a) Find a generalform equation of the plane trl that contains 11 and is parallel to 12. Answer: A: 17y 132 2 73 (b) Find a generalform equation of the plane 7r2 that is perpendicular to trl and contains 12 Answer: C] (c) Find the point of intersection of the plane T72 and the line 11 Answer: C] Can you see the geometric significance of this point? Consider the line Ht) = (3, 1, 1)i t (1, 3, 2). (a) The point PU], 8, 7) is on i and corresponds to the parameter value t l3 1 (b) The point Q( 8,16, 7) m on 1. (c) The point R( 26, 28, 38) is not on 1 The distance between R and l is C] and the point on I closest to R is C]. (1 point) Find an equation for the plane that contains the line a: = 6 l 5t ll : y : t 7 and is perpendicular to the z = 3 213 plane -59: + y + 52 : 55. Plane equation in general form: (1 point) Find the point of intersection of the two lines l1 : if = (~16,2,16) + 15(H333, 2) and 12:5? : (5,1.9,24} + t (~5, 5, 4) Intersection point: (1 point) Let ll : i5 : (5, 3, 4) + t (20, 16, 20) and a: :e4e25t lg: 3} =20: z :5+25t (a) What is the geometric relation between the Hnes? The lines are ? C (b) Find an equation (in general form) for the plane containing both lines. If there is no such plane, or if there are infinitely many, enter "impossible". Plane equation