23 Vectors 3 BMQ Soln (RVHS)

1 Tutorial 8C: Vectors III (Lines in Three-Dimensional Space) Basic Mastery Questions 1. Find a vector equation for each of the following lines: (a) a line which passes through the point with coordinates (1, 2, 3) and is parallel to the vector 3i  2j + 4k; (b) a line which passes through the points (2, 0, 6) and (3, 5, 1). (a) 1 3 : 2 2 3 4 r                   l , where  (b) 2 3 5 1 0 5 5 5 1 6 1 5 1                                         Ans: 2 1 : 0 1 6 1 r                   l , where  2. Convert the following vector equations to Cartesian equations. (i) 2 1 3 1 1 2                    r ,  (ii)               5 0 3 0 4 0 r , . (i) 12 3 2    zx y (ii) , 43 5 x z y 5 3 , 4  x z y 3. Convert the following Cartesian equations of straight lines to vector equations. (i) 155 1  zyx (ii) yx  2 12 , 5z (i) 1 5 0 5 1 1 r                  , where  (ii) 1 12 0 1 5 0 r                   , where 

2 4. (N07/1/7) The point P is the foot of the perpendicular from the point A 1,3, 2 to the line given by 3 8 3 2 2 3    x y z . Find the coordinates of P, and hence find the length of AP. [solution] Vector equation of this line is 3 2 8 2 3 3 r                  , where  Let B be the point 3,8,3 which lies on the line. 1 3 4 3 8 5 2 3 5                                 BA   3 4 2 2 1 18 5 2 2 17 173 5 3 3 3 28 10 158 2 173 3 5 6 0 m m m m                                                                                        BP BA OP OB BA ( 5,6,0) P 5 1 6 Hence, 6 3 3 36 9 4 7 units 0 2 2                                       AP OP OA B(-3, 8, 3) P A (1, 3, -2) 2 2 3 m      