scrimmage 3 DE ANZA COLLEGE NAME MATH 43 03/20/17 E. Njinimbam YOU MUST SHOW YOUR WORK CLEARLY AND LOGICALLY FOR CREDIT. 1.) For the conic section with equation r = 3 . 4 + 5sin a.) Find the eccentricity , e. b.) Find the equation of thedirectrix . c.) Identify the type of conic section, and find its vertices. 2.) a.) Find the parametric equation of the line segment from A( 4,3) to B(2,2) 1 b.) Find the coordinates of the point 3 of the way from B to A. 27 3 3.) If u v , then proju v = 0 . Find, exactly, projv ( u v ) if v = 21.34569, 3.789 , ,and 13 u = 21, - 7.394, 2 . 4.) x = et 1 Remember to show orientation. y = 3 + e 2t [Hint: Change the equation to rectangular coordinates ] Sketch the graph of : 5.) 6.) Find the parametric equation of the circle of radius 4 centered at (4 , 3), traced clockwise starting at the point (8 , 3) when t = 0. a.) Change to polar form : P(-4 , -4) 5 & $ b.) Change to rectangular form : P 6, % 6' c.) Change to rectangular form : r 2 = 3sec2 d.) Change to polar form: x 2 + y 2 = 4 y 1 # l1 : x = 1 + 4t ; y = 5 4t ; z = 1 + 5t Find the distance between the lines $ . Lines are skew. % l2 : x = 2 + 8t ; y = 4 3t ; z = 5 + t Note: Pi li , i = 1, 2, with P1P2 = 1, 1, 6 v 2 = 8, 3,1 = 17, 47, 5 , P1P2 v2 = 29 3 , v2 = 74 , P1P2 v2 The distance, d = v2 8.) Find the equation of the sphere with center (2, 1,- 3) that is tangent to the plane x - 3y + 2z = 4 ( Be thorough in your explanation or work ) n P1Po Note: The radius, r = projn P1Po = , P1(2,1,-3), Po(4,0, 0), P1P0 = 2, 1, 3 , n = 1, 3, 2 n n P1Po = 11 , and n = 14 7.) Find the vector component of u = 1,- 2, 4 orthogonal to v = 0, 2, 3 . u iv Note: u iv = 8, and v = 13 , and projv u = 2 v , and n = u projv u , where n is the required vector. v 9.) 10.) Set up the system of equations. Do not solve: A load of 300N at the origin is supported by three cables hung at A(0,10, 10), B(-4,-6,10), and C(4,-6,10) respectively. Find the tension in each of the supporting cables. = 1 0,10,10 , OB = 1 4, 6,10 , OC = 1 4, 6,10 , w = 0,0,300 Let OA 2 38 2 38 10 2 , OB = OB OB , OC = OC OC , and for equilibrium OA + OB + OC + w = 0 = OA = OA OA 0,0,0 Now equate components to get the equations 2 # % Determine the point of intersection of the lines 1 + t, 3 + t, 1 t and (2t, 3 t, t ) . $ & 3 Note: vi li , i = 1, 2, with v1 = 1, 2 / 3, 1 k 2, 1,1 =k v 2 , for any k . So l1 l2. Change one of the parameters to s, then equate the corresponding coordinates of the lines and solve for t, and s. substitute lines to get therequired point. the values of t, and s in their respective 11.) 12.) Locate the point of intersection of the plane 2x + y - z = 0 and the line through (3, 1, 0) that is perpendicular to the plane. ( Be thorough in your explanation or work ) Note: let n = 2,1, 1 plane , and n line , line, l : (3+2t,1+t,-t). Substitute the coordinates of l into the equation of the plane and solve for t. Use this value in l to get the required point. 13.) Find an equation of the plane that passes through the line of intersection of the planes x - z = 1 and y + 2z = 3 and is perpendicular to the plane x + y - 2z = 1 ( Be thorough in your explanation or work ) Note: Let n1 = 1, 0, 1 , n2 = 0,1, 2 , n3 = 1,1, 2 , ni plane i . Let Pl(1,3,0) line. Let v = n1 n2 . Let n = v n3 , P(x,y,z). Then the equation of the plane is given by: n Pl P = 0 2 14.) Find an equation for the plane consisting of all points that are equidistant from the two points A(1,1,0) and B(0,1,1) . Let Pm be the midpoint of A(1,1 0) and B(0,1,1) Note: let P(x,y,z), and BA plane . Equation of the plane is given by: BA Pm P = 0 15.) Find the equation of the plane through (1,-9,4) and perpendicular to the planes 5x + 3y - z - 1 = 0, and x - y + 2z - 8 = 0 . ( Be thorough in your explanation or work ) Note: Let n1 = 5, 3, 1 , n2 = 1, 1, 2 , ni plane i . Let Pl(1,-9,4), and P(x,y,z). Let n = n1 n2 . Then the equation of the plane is given by: n P1P = 0 2 The position vectors of two bugs at time t are 3(t -1) i + 9(t - 1) j and t 6 i + 2t 3 j . Show that the bugs never collide, but their paths cross twice. Note: Change one of the parameters to s, then equate the corresponding coordinates and solve for t and s. Use the values of t and s to find the points of intersection. 16.) 17.) Find the equation of the plane that contains the point (2,1,1) and the line (2 t, 1 + 4t, 1 + 4t ) . Note: let P1(-2 , 1, 1), P2(2 , 1 , 1), and v = 1,4,4 , then the normal n = v x P1P2 . let P(x,y,z), Then the equation of the plane is given by: n P1P = 0 3