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 Page 3 of 6 7 The equation of two lines and a plane are as follows: l 1 : r = i + 2j + 3k +  (i + 3j – 2k),   I R l 2 : r = i + 2j + 3k +  (mi + 5j – k),   I R  : r  (3i  2j + k) = 10 (i) Find the angle between the line l 1 and the plane . Ans: 30 (ii) Find the possible value(s) of m if the angle between the line l 2 and the plane  is 60. Ans: 4 or 32 Optional Question: It is given that a = a1i  a2j  a3k and b = b1i  b2j  b3k. (a) By considering the scalar product a  b, prove that (a1b1  a2b2  a3b3)2  (a12  a22  a32) (b12  b22  b32). (b) By considering the vector product a  b, prove that (a1b2  a2b1)2  (a2b3  a3b2)2  (a3b1  a1b3)2  (a12  a22  a32) (b12  b22  b32). 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 4 of 6 Homework C: Vector Product and Planes – Part 2 1. For each of the following pairs of planes, find the acute angle between the planes and an equation of their line of intersection. (i) r              4 3 9 = 7 and r              1 1 2 = 1 (ii) r             2 3 1 = –4 and r            1 1 1 = 1 2. Show that A(2, 3, –2) lies in the plane 1π and 2π whose equations are r              2 2 3 = 4 and r            1 3 1 = 9 respectively. Find a vector parallel to both 1π and 2π . Hence, deduce an equation for the line where 1π meets 2π . 3. For each of the following pairs of planes, find an equation of their line of intersection. (i) r = 2i + j +  (– j – k) + (i – j – k) ,  ,   I R and r = 2j – k + s (2i – j) + t(i – k) , s , t  I R (ii) r = 3i + k +  (2i – j + 2k) + (2i + 3j + 2k) ,  ,  I R and r             1 6 2 = –5 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 5 of 6 4. For each of the following, determine if the three planes intersect and find the point or line of intersection when it exists. (i) 1523  zyx , 1032  zyx and 72  zyx (ii) 62  zyx , 15242  zyx and 82  zyx (iii) 222  zyx , 6256  zyx and 4 zyx (iv) 222  zyx , 222  zyx and 8368  zyx 5. The equations of planes 1π , 2π and 3π are 625  zyx , 432  zyx and azyx  4 respectively. Discuss the solution of these three equations when (a) 9a (b) 2a giving a geometrical interpretation for each case. Answers: 1. (i) 6.7 , r =                       3 1 1 4 3 0 λ ,  I R (ii) 9.51 , r =                      4 3 1 1 2 0 λ ,  I R 2.            11 5 4 , r =                       11 5 4 2 3 2 λ ,  I R 3. (i) r =                        1 1 3 1 0 4 λ ,  I R (ii) r = 12 01 32                   ,  I R 4. (i) (4, 1, 1) (ii) No intersection (iii) No intersection (iv) r =                      2 1 0 0 0 1 λ ,  I R 5. (i) No common point; each plane is parallel to the line of intersection of the other two planes. (ii) They intersect at the line r =                        3 11 1 0 3/8 3/2 λ ,  I R 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 6 of 6 Quiz The lines L1 and L2 meet at the point P. The line L3 is coplanar with L1 and L2 and is perpendicular to L1. Given that L1 and L2 are parallel to the vectors a and b respectively, show that L3 is parallel to the vector   abba a 2 . The equations of L1 and L2 are now known to be                    3 11 5 10 22 tr and                    3 17 53 24 sr respectively, where s and t are real parameters. Find the equation of the line L3, given that L3 also passes through P. The line L4 has equation                     10 3 34 11 ur , where u is a real parameter. Determine if L3 and L4 are skew or intersecting. The line L5 is perpendicular to both L3 and L4. Find the acute angle between L5 and the plane containing L1 and L2. Ans:                      36 57 22 r ; L3 and L4 are skew lines ; 83.9 This study source was downloaded by 100000857398866 from CourseHero.com on 11-19-2024 23:34:4