NUS High School Adv Math Homework Worksheet Vectors

MA6132 Advanced Mathematics II Page 1 of 6 Name: ______________________________ ( ) Date: ________________ Homework C: Vector Product and Planes – Part 1 1 The position vectors of points A, B and C are a = i  j, b = i  k and c = j  k respectively. Find (i) a  b, Ans: i – j – k (ii) b  c, Ans: – i – j  k (iii) (a  b)  c, Ans: – j  k (iv) a  (b  c). Ans: i – j 2 Relative to the origin O, the position vectors of points A and B are OA  = i  2j  2k and OB  = 2i  3j  6k respectively. The point P on AB  is such that AP : PB =  : 1  . Show that OP  = (1  )i  (2  5)j  (2  8)k. (a) Find the value of  if OP  is perpendicular to AB . Ans: 5 18 (b) Find the value of  if AOP = POB. Ans: 3 10 3 The position vectors of points A and B are a = 9i  9j  9k and b = 7i  14j  14k respectively. (a) Find the length of projection of a on b. Ans: 3 (b) Use your answer in (a) to resolve the vector a into two components, one parallel to b and one perpendicular to b. Ans: i – 2j + 2k, 8i + 11j + 7 This study source was downloaded by 100000857398866 from CourseHero.com on 11-19-2024 23:34:49 GMT -06:00 https://www.coursehero.com/file/220718965/HomeworkC-1pdf/

Homework C: Vector Product and Planes Page 2 of 6 4 Referred to the origin O, the position vectors of points A, B , C and D are i  j, i  k, i  j  k and j  2k respectively. Find a (i) parametric equation for the plane containing the points A, B and C, Ans: r = i  j  (j  k)  k, ,   I R (ii) vector (scalar product) equation for the plane containing the points A, B and D, Ans: r  (j  k) = 1 (iii) Cartesian equation for the plane containing the points A, C and D. Ans: 2x  y = 1 5 The vector equation of a plane  is r  (i  2j  2k) = 1. (i) Show that the point A(9, 2, 6) lies in the plane . (ii) Show that the point B(2, 5, 6) does not lie in the plane , and find the shortest distance from the point B to the plane . Ans: 7 (iii) The shortest distance from the point C(2009, 7, m) to the plane  is 26. Find the possible value(s) of m. Ans: 972 or 1050 (iv) Find the coordinates of the foot of perpendicular from the point D(4, 7, 5) to the plane . Ans: (1, 1, 1) 6 The vector equations of a line l and a plane  are r = mi – j +  (2i – 2j + k),   I R and r  (i  2j + 2k) = 1 respectively. (i) If m = 3, show that the line l lies in the plane . (ii) If m = 6, find the shortest distance from the line l to the plane . Ans: 1 (iii) Find the possible value(s) of m if the shortest distance from the line l to the plane  is 7. Ans: 18 or 24 This study source was downloaded by 100000857398866 from CourseHero.com on 11-19-2024 23:34:49 GMT -06:00 https://www.coursehero.com/file/220718965/HomeworkC-1pdf/

Homework C: Vector Product and Planes