Math 150B Exam 4 Solution

18 Exam (4) Math 150B. Calculus II Section # 18056 Fall 2022 Instructor: Henrik Sarkssian THURSDAY DECEMBER 8, 12:30 PM - 13:35 PM Name: Nyi Nyi Moc Hlet (Please print) - Show all of your work and make the final answers clear. Full credit will NOT be given for merely stating the right answer without any appropriate justification. - Cell phones must not be visible or audible at any time during the exam. This applies to all smart technology including watches, cameras, and pens. No Calculators! Problem 1 2 3 4 5 6 Total Maximum [25] [25] [25] [25] [25] [25] 150 Score 25 25 25 25 25 25 150 Awesome! 1

Problem 1. (25 pts.) Consider a triangle with vertices: A(-2,5,3),B(5,3,4),(-3,1,2). Find all angles of the triangle. 8 AB = (5+2,3-5,4-3) -( 30 AC = = IABI = 49 + 4 + 1 = 54 IACI / 1 16+1 /18 cos ( BAC F AB . Ac = . 0 LBAC = 1=90 IAB/1ACI /54 / 18 2 A=

Problem 2. (25 pts.) Consider a triangle with vertices: A(3,-5,2),B(5,-4,4),C(-1,-2,-3) (i) Find the area of the triangle. (ii) Find two unit vectors perpendicular to the plane of triangle ABC. B (i) = D C A AB x AC = ijk = (-5-6)i-(-10+8))+(6+4)k 212 -43-5 I AB X AC I = / 121 . 4 + 100 = = /225 = 15 = (ii) x = First Unit Vector Second Unit Vector 25 3

Problem 3. (25 pts.) For the vectors u= (2,-4,6) and v=(2,-3,1) express u as the sum u =p+n, = where p is parallel to V and n is orthogonal tov. Bajo is = J.V XX P V (4+12+6)(22-3,1) 4+9+1 = = - 25 4

Problem 4. (25 pts.) Show that the following trajectory lies on a sphere in R3. Find the radius of the sphere, and show that the position vector and the velocity vector are everywhere orthogonal. r(t) = + 2 superscript(2) = sin 2t + 4 cos 2t + 3 sin22t = 4 sin 2t + 4 cos'21 = ( sin' 2t + cos' 2t) = 4 =22 Sphere Radius = 2 2t, -4 sin 2t, 2 13 cos 2t ) (t)..r'(t) = 2 sin2t cos 2t - 8 sin 2t cos 2t + 6 sin 2t cos 2t = 8 sin 2t cos 2t - 8 sin 2t ws 2t =0 Therefore, pesition vector and velocity vector are everywhere orthogonal. , 25 5

Problem 5. (25 pts.) Find the length of the following curve on the given intervals. r(t)= t - cost, t+ cost, V2sint); 05151 sint, 1- - sint, 52 cost> = sin t) + = 1 + 2 sint + sin't + 1- 2 sint + sin't + 2 cos't = = /2+2 (sin't+as't) = =2 L- P! dt. P' 2 dt = 2t !! = 12-0=2 25 6

Problem 6. (25 pts.) Given an acceleration vector, initial velocity and initial position (xo,yo,zo) find the velocity and position vector for . 4(0)+C1 00 0+C230+C3=6 G = 0 C2 33 C3=6 = (t) - Svcts dt = < 2t'+C4 , " 1/1 2(0)++4 49c53 C6 21 r = 15