RI Vectors C2B Tut (Qn)

RAFFLES INSTITUTION H2 Mathematics (9758) 2025 Year 5 ___________________ Tutorial 2B: Vectors II Page 1 of 5 Tutorial 2B: Vectors II – Equations of Straight Lines Section A (Basic Questions) 1 For each of the following, write down a vector equation of the line l and convert it to Cartesian form. (a) l passes through the point with position vector 2ij k  and is parallel to the vector ij . (b) l passes through the points (1, 1, 3)P  and (2,1 , 2)Q  . (c) l passes through the origin O and is parallel to the line 11 :1 2 , 33 m              r  . (d) l passes through the point (1,0,1)C and is parallel to the y axis. [(a) 1 2, 1x yz  (b) 131 25 yzx   (c) 23 yzx (d) 1, 1x z ] 2 For each of the following, find the acute angle between the lines 1l and 2l . Determine if 1l and 2l are parallel, intersecting or skew. In the case of intersecting lines, find the position vector of the point of intersection. (a) 1 :1 2lx y z  and 2 21 3: 22 2 x yzl    (b) 1 14 :0 2 , 03 l               r  and 2 03 :1 0 8 , 11 l             r  (c) 1 :( 5 )( ) , l      rik i j k  and 2 :( )( 5 4 ) , l     ri j k ij k  [(a) 0 , parallel (b) 81.3 , skew (c) 44.5, intersecting, 6 5 0   ]

Raffles Institution H2 Mathematics 2025 Year 5 ____________________________________ _____ ___________________ Tutorial 2B: Vectors II Page 2 of 5 Section B (Discussion Questions) 1 The points P and Q have coordinates ( 0 ,1 ,1 ) and (3,0,1) respectively, and the equations of the lines 1l and 2l are given by 1 00 :1 1 , 31 l         r  and 2 32 :3 1 , 10 l          r  . (a) (i) Show that P lies on 1l but does not lie on 2l . (ii) Determine whether the lines 1l and 2l passes through Q . (b) (i) Find the coordinates of the foot of perpendicular from P to 2l . Hence, or otherwise, find the perpendicular distance from P to 2l . (ii) Using (b)(i) or otherwise, find the length of projection of PQ   onto 2l . [(a)(ii) Q is on 2l but not 1l (b)(i) (1,1,1) , 3 units (b)(ii) 5 units] 2 The lines l and m are defined by the equations :( 2 6 3 ) , 13:. 44 l xa y zm a     ri k i j k (i) Given that the lines intersect, show that 6a . [2] (ii) Find the position vector of N, the foot of perpendicular from the point (5, 0,1)A to the line l. [3] (iii) Hence or otherwise, find the position vector of the two points on l that are 5 units from A. [3] [(ii) 11 1 127 1   (iii) 17 1 307 8   , 5 1 67 10    ] 3 The points A and B have position vectors 2j k and 22ij k respectively. (i) Find a vector equ