MAC 2313LINES AND PLANESExercise 1. You are given vectors ~a = h1, −2, 2i and~b = h2, 0, −3i.For what value of t does the vector ~a + t~b lie in the plane determinedby the vectors ~p = h1, 1, 1i and ~q = h1, 2, 3i?Hint: Try to use triple scalar product. The final answer is t = 7.Exercise 2. Use properties of the cross product and triple scalarproduct to simplify the following expressions:(i) (~a ×~b) × (~b ×~a) (ii) (~a ×~b) ·~aExercise 3. You are given the points A(1, 5, −1), B(3, 2, −2) andC(1, 3, a), where a is a real number. Let l be the line determined byA and the vector ~s = h1, −1, 3i. Find the value of a so that the planedetermined by A, B and C contains the line l.Answer: a = −15.Exercise 4. Consider the linel :x − 43=y + 3−4=z − 32.Find an equation of the line that passes through the point (3, 2, 0) andis perpendicular to l.Hint: First, find an equation of the plane containing the point (3, 2, 0)and is perpendicular to l. Use this plane to get the second point onthe line p erpendicular to l. The final answer isx−34=y−2−9=z−05.Exercise 5. Let l1be defined as the line of intersection of the planesy = 3 and 2x + y − z = 6. Additionally, let the line l2be defined asl2:x − 12=y − 12=z − 21.Show that the lines l1and l2belong to the same plane.Hint: You need to show that l1and l2intersect at some point or thatl1and l2are parallel. In other words, you need to show that l1and l2are not skew lines.12Exercise 6. You are given the planesπ1: 2x − y + z = 7, π2: x − y = 4.Let m be the line defined as the intersection of the planes π1and π2.You are also given the linel :x − 20=y + 11=z + 14.Let T1and T2be the points of intersection of l with the planes π1andπ2respectively. Find the projections of T1and T2onto m.Hint: First, you need to find T1and T2. You should get T1(2, 0, 3)and T2(2, −2, 5). Then, you need to find the line m, and you will getm :x−41=y−01=z+1−1. The projection of T1onto m is the intersectionof m and the plane through T1perpendicular to m. The same holdsfor T2. The final answer is as follows: the projections of T1and T2ontom are (2, −2, 1) and (4, 0, −1) respectively.Exercise 7. Consider the plane π : x −3y + 2z + 5 = 0 and the sphereof radius 12 centered at (5, −14 , 9).(i) Find the point P on π with the smallest distance from the sphere.(ii) How far is P from the center of the sphere?(iii) How far is P from the sphere?Answer: (i) P (0, 1, −1); (ii) 5√14; (iii) 5√14 − 12.Exercise 8. Let S be the sphere with the following properties: theradius of S is 3, the sphere S touches the plane 6x + 3y + 6z = 9 at thepoint (1, −1, 1), and S is located in the half-space 6 x + 3y + 6z ≥ 9.(i) Write an equation of the sphere S.(ii) Find all intersection p oints of the sphere S with the plane y = 3.Answer: (i) (x − 3)2+ y2+ (z − 3)2= 9; (ii) point (3, 3,