RI 2024 RTT2 Vectors(Worksheet)

RAFFLES INSTITUTION H2 Mathematics (9758) 2024 Year 5 ________________________________________ Y5 H2 Math RTT2 Focus Lesson on Vectors Page 1 of 4 Term 4 RTT2: Post-Promo Revision Lesson Worksheet Session 3: C4A to C4C: Vectors 1 Referred to the origin O, the points A and B have position vectors a and b respectively. The vectors a and b are given by 263a n d 2 ,ppp   aij k b i j k where p is a constant. Find ab a and give a geometrical interpretation of ab a . [2] Find ab and give a geometrical interpretation of ab . [2] Given that ais a unit vector, find the possible value(s) of p . [2] 2 DHS Prelim 9758/2020/01/Q6b modified With reference to the origin O, the points A and B are such that OA   a and OB   b. It is given that 2 3OX   a, 3 4OY   b and the line ON bisects the line XY at the point M. (i) By considering the ratio :XM MY , find the vector OM  in terms of a and b. [1] (ii) Given that :: 1AN NB  and :: 1ON OM k  where  and k are real constants, find the ratio AN : NB. [4] 3 (a) The non-zero vectors a, b and c are such that ab ca . Given that bc , find a linear relationship between a, b and c. [3] (b) The variable vector vij kabc satisfies the equation  3.vi k j  Find the set of vectors v and describe this set geometrically. [3] A B O X N Y M

Raffles Institution H2 Mathematics 2024 Year 5 __________________________________________ Y5 H2 Math RTT2 Focus Lesson on Vectors Page 2 of 4 4 ACJC Promo 9758/2020/Q8 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) Find the position vector of the two points on l that are 5 units from A. [3] 5 9740/2015/02/Q2 The line L has equation 24 ( 236 ) .    ri j k i j k (i) Find the acute angle between L and the x-axis. [2] The point P has position vector 256 .ij k (ii) Find the points on L which are a distance of (33) from P. Hence or otherwise find the point on L which is closest to P. [5] (iii) Find a cartesian equation of the plane that includes the line L and the point P . [3] 6 JPJC Promo 9758/2020/Q9 modified The plane 1p has equation 10 2 1 , where and are real parameters. 01            r . (i) Find an equation of 1p in the form dr.n . [3] The plane 2p has equation 21 2xyz . (ii) State the relationship between 1p and 2p . [1] The line l has equation 3 2 2 t    r , where t is a real parameter. (iii) Find the acute angle between l and 1p . [2] (iv) Find the foot of perpendicular from the origin to 2p . Hence, or otherwise, find the exact distance between 1p and 2p . [4]

Raffles Institution H2 Mathematics 2024 Year 5 __________________________________________ Y5 H2 Math RTT2 Focus Lesson on Vectors Page 3 of 4 More Practice Questions 7 MI PU3 Mid-Year CT 9758/2018/01/Q3 Relative to the origin O, two points A and B have position vectors a and b respectively. It is given that b is a unit vector, 3, and 4 3 41. aa b  is defined as the acute angle between a and b . (i) By considering the scalar product (4 3 ) (4 3 ),ab ab  find . [4] (ii) Give the geometrical meaning of () ab b and find its exact value. [2] 8 9758/2019/02/Q5 With reference to the origin O, the points A, B, C and D are such that OA a  , OB b  , 24OC  ab   and 5ODba   . The lines BD and AC cross at X (see diagram). (i) Express OX  in terms of a and b . [4] The point Y lies on CD and is such that the points O, X and Y are collinear. (ii) Express OY   in terms of a and b and find the ratio :OX OY . [6]

Raffles Institution H2 Mathematics 2024 Year 5 __________________________________________ Y5 H2 Math RTT2 Focus Lesson on Vectors Page 4 of 4 9 9758/2019/01/Q12 A ray of light passes from air into a material made into a rectangular prism. The ray of light is sent in direction 2 3 6   from a light source at the point P with coordinates (2,2,4). The prism is placed so that the ray of light passes through the prism, entering at the point Q and emerging at the point R and is picked up by a sensor at point S with coordinates  5, 6, 7 . The acute angle between PQ and the normal to the top of the prism at Q is  and the acute angle between QR and the same normal is  (see diagram). It is given that the top of the prism is a part of the plane 1,xyz and that the base of the prism is a part of the plane 9xyz  . It is also given that the ray of light along PQ is parallel to the ray of light along RS so that P, Q, R and S lie in the same plane. (i) Find the exact coordinates of Q and R. [5] (ii) Find the values of cos and cos . [3] (iii) Find the thickness of the prism measured in the direction of the normal at Q. [3] Snell’s law states that sin sin k , where k is a constant called the refractive index. (iv) Find k for the material of this prism. [1] (v) What can be said about the value of k for a material for which  ? [1]