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 equation of the line l passing through the midpoint, M , of AB and the origin O . (ii) Find the position vector of a point N where N is the foot of the perpendicular from A to l . (iii) Find the position vector of a point C such that AOCM is a parallelogram. (iv) Show that the points , A N and C are collinear. State, with a geometrical reason, the value of CO CM . [(i) 1 :1 , 0 lt t   r  (ii) 1 1 12 0 ON      (iii) 1 0 2 OC     (iv) 1]

Raffles Institution H2 Mathematics 2025 Year 5 ____________________________________ _____ ___________________ Tutorial 2B: Vectors II Page 3 of 5 4 In the diagram, O is centre of the rectangular base ABCD of a right pyramid with vertex V . Perpendicular unit vectors , ,ijk are parallel to ,AB BC and OV respectively. The length of AB , BC and OV are 12 cm, 6 cm and 6 cm respectively. A line l has cartesian equation 2210 4   ztyx . (i) Find the vector equation of line AV . [2] (ii) If the line l intersects line AV at M , find the position vector of M and the value of t . [5] (iii) Find the acute angle between line AV and the line l . Hence find the perpendicular distance from A to the line l . [5] [(i) 02 01 , 62          r  (ii) 4 2 2 OM    , 2t  (iii) 60.8 , 2.62] 5 Referred to the origin O, points A and B have position vectors a and b respectively. Point C lies on OA, between O and A, such that OC : CA = 3 : 2. Point D lies on OB, between O and B, such that OD : DB = 5 : 6. (i) Find the position vectors OC   and OD   , giving your answers in terms of a and b. [2] (ii) Show that the vector equation of the line BC can be written as 3 1,5r= a b where  is a parameter. Find in a similar form the vector equation of the line AD in terms of a parameter . [3] (iii) Find, in terms of a and b, the position vector of the point E where the lines BC and AD meet and find the ratio AE : ED. [5] [(i) 35 ,51 1ab ( i i ) 5 111 r= b a (iii) 91 20 4a+ b, 11 : 9 ] D V C B 12 cm 6 cm O k j i 6 cm A

Raffles Institution H2 Mathematics 2025 Year 5 ____________________________________ _____ ___________________ Tutorial 2B: Vectors II Page 4 of 5 6 When referred to the origin O, the points A and B have position vectors 51 5j k and 3ij k respectively. The line 1l has equation 21 12 , 33          r  . (i) Find a vector equation of line 2l passing through the points A and B. [2] (ii) Find the coordinates of the point C, where 1l and 2l intersect. [3] (iii) Find the position vector of the point F, the foot of the perpendicular from A to the line 1l . [4] (iv) Find the vector equation of the line of reflection of 2l in the line 1l . [3] [(i) 01 56 , 15 12                r (ii) (2, 7, 9) (iii) 5 7 12    , (iv) 26 78 , 99 r            ] 7 One day, Eddie came home fr om a birthday party and brought back a helium filled balloon. After playing with it, he accidentally released the balloon at the point (1, 2, 3) and it floated vertically upwards (in a path parallel to the vector k) at a speed of 1 unit per second. t seconds later, a sudden gust of wind ca used the balloon to move in the direction of i  4j 6k. (i) Find the angle in which the balloon has cha nged in direction after the gust of wind blew it away. [3] (ii) Given that the balloon eventually stayed at the point (2, 6, 12) on the ceiling, find the time t when the gust of wind blew the balloon away. [3] Eddie decides to shoot the balloon down with his catapult. (iii) Assume he was holding his catapult at (3, 2, 1) initially and he walked along the path parallel to 2 i  j. Find the position vector of the point where he should place his catapult so that the distance between his catapult and the balloon is at its minimum. Hence find this distance. [4] [(i) 34.5o (ii) 3t  (iii) 19 1 125 5    , 11.7]

Raffles Institution H2 Mathematics 2025 Year 5 ____________________________________ _____ ___ ________________ Tutorial 2B: Vectors II Page 5 of 5 8 A frigate is stationed at position  1, 2, 0F . Two submarines 1S and 2S are under the sea surface. Submarine 1S is at position  2, 1, 1A  and travelling in a path parallel to vector –3i + 2j – k. An enemy submarine 2S is detected at position  3, 2, 2B  travelling in a path parallel to vector –2i – 3j + k. (i) Determine if the paths of the submarines will intersect each other. [3] (ii) The enemy submarine 2S will launch a torpedo at the frigate when it is at a