Updated 2021 Ch 12 Vectors II Tutorial (Student)

2021 SAJC JC1 H2 Mathematics Tutorial Chapter 12: Vectors II Page 1 Tutorial 12 : Vectors II (Planes) Practice Questions 1. The plane is given by the Cartesian equation 3x + y – 2z = 3. (a) Find the perpendicular distance of the plane from the origin. (b) Find the perpendicular distance of the point (1, 3, – 1) from the plane. [Ans: (a) 14 3 , (b) 5 14 ] N2007/P1/Q8 2. The line l passes through A and B with coordinates (1, 2, 4) and (-2, 3, 1) respectively. The plane p has equation 3x – y + 2z = 17. Find (i) the coordinates of the point of intersection of l and p, (ii) acute angle between l and p, (iii) the perpendicular distance from A to p. [Ans: (i)  2.5,1.5,5.5 (ii) 78.8o (iii) 4 14 7 ] 3. N2009/P1/Q10 The planes 1p and 2p have equations r 2 1 3    = 1 and r 1 2 1    = 2 respectively, and meet in a line l. (i) Find the acute angle between 1p and 2p . [3] (ii) Find a vector equation of l. [4] (iii) The plane 3p has equation 2x + y + 3z – 1 + k(-x + 2y + z – 2) = 0. Explain why l lies in 3p for any constant k. Hence, or otherwise, find a Cartesian equation of the plane in which both l and the point (2, 3, 4) lie. [5] [Ans: (i) 70.9o (ii) r = 01 1 1 , 01                        (iii) x – y = -1] 4. N2013/P2/Q4 The planes 1p and 2p have equations r. 2 2 1   = 1 and r. 6 3 2    = - 1 respectively, and meet in the line l. (i) Find the acute angle between 1p and 2p . [3] (ii) Find a vector equation for l. [4]

2021 SAJC JC1 H2 Mathematics Tutorial Chapter 12: Vectors II Page 2 (iii) The point A (4, 3, c) is equidistant from the planes 1p and 2p . Calculate the two possible values of c. [6] [Ans: (i) 40.4 o (ii) r = 1/ 6 2 / 3 , 01 7 / 6 5 / 3                       (iii) 35 13 or – 49] 5. N2014/P1/Q9 Planes p and q are perpendicular. Plane p has equation x + 2y – 3z = 12. Plane q contains the line l with equation 1 1 3 2 1 4 x y z    . The point A on l has coordinates (1, -1, 3). (i) Find a Cartesian equation of q. (ii) Find a vector equation of the line m where p and q meet. (iii) B is a general point on m. Find an expression for the square of the distance AB. Hence, or otherwise, find the coordinates of the point on m which is nearest to A. [Ans: (i) x – 2y – z = 0 (ii) r = 64 3 1 , 02                    (iii) 2 2 21 36 50AB    ; 6 3 5 7 4    ] 6. The equations of the line l1 and plane  1 are as follows: l1 : 51 1 1 ,~ 40 r                       ;  1 : 45  azxa , where a is a positive constant. (i) If the angle between l1 and  1 is 6  , show that 1a . (ii) Find the position vector of A, the point of intersection between l1 and  1. (iii) Given that C(7,3,4), find the position vector of N, the foot of perpendicular of C on  1. (iv) Point 'C is obtained by reflecting C about  1. Determine the vector equation of the line 'AC . [Answers: (ii) 54i j k (iii) 6 3 3i j k (iv) 5 1 41 0 1                  r ]

2021 SAJC JC1 H2 Mathematics Tutorial Chapter 12: Vectors II Page 3 7 [NYJC/2018/Prelims/P1/Q5] The line l passes through the points A and B with coordinates (5, 2, 4) and (4, 1, 3) respectively. The plane p has equation 4 7 5 24.x y z   (i) The point C lies on l such that the foot of perpendicular of C onto p has coordinates (3, 1, 1). Find the coordinates of C. [4] Plane 1p has equation 3 2 .x y z    (ii) What can be said about the values of  and  if l does not intersect 1 ?p [2] (iii) Hence find the exact values of  if the distance between 1p and l is 2 units. [3] [Ans: (i) (7, 8, 6) (ii) 3  , 23  (iii) 23 2 22 or 23 2 22    ] 8. RHVS/2018/Prelim/P1/Q4 Relative to the origin O , the points A , B and C are such that OA    a , OB    b and OC    c respectively, where a , b and c are vectors which are mutually non-parallel. The plane  and the line l have the following equations : where ,      r a b and : where l   r a c respectively. (i) Find the point(s) o f intersection between  and l given that the points O , A , B and C are (a) coplanar, (b) not coplanar. [3] (ii) State the geometrical meaning of 1 2 ab . [1] In the rest of the question, O, A, B and C are not coplanar. (iii) The vector p is a unit vector in the direction of ab . State the geometrical meaning of cp . [1] (iv) It is given that 1 1 0 OA     , 0 1 1 OB     and 0 0 2 OC     . Find the volu me of the pyramid OABC . [3]

2021 SAJC JC1 H2 Mathematics Tutorial Chapter 12: Vectors II Page 4 [Ans: (iv) 21 units3 ] 9. TPJC Prelim 9758/2018/02/Q3 The line l has equation 95 ,131 xy z  , and the plane 1p has equation 26x y z    . (i) Find the acute angle between l and 1p . [3] Referred to the origin O, the point A has position vector 26i j k . (ii)Find the position vector of F, the foot of the perpendicular from A to 1p . [3] (iii) Find the perpendicular distance from A to 1p , in exact form. [2] (iv) Given that l is the line of intersection of the planes 2p and 3p with equation 3x y z a   and 7x by z   respectively, where a and b are real constants, find the values of a and b. [4] [Ans: (i) 7.4  (ii) 0 5 4 OF   (iii) 6 (iv) 5, 3ab  ] 10. 2019 A levels Exams/P1/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

2021 SAJC JC1 H2 Mathematics Tutorial Chapter 12: Vectors II Page 5 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 1x y z   , and that the base of the prism is a part of the plane 9x y z   . 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] [Ans: (i) 8 1 2,,11 11 11Q  , 37 39 23,,11 11 11R   (ii) 11 3cos 21  , 11 510cos 255  (iii) 10 3 3 (iv) 1.86k  or 170 7k  ] 11. CJC Prelim 9758/2020/01/Q11 A temporary isolation centre is built to manage the increasing number of COVID-19 cases. The roof takes the shape of a triangular prism. Points  ,,x y z are defined relative to an origin, O, with unit vectors i along OA , j along OC , and k along OD (see diagram). The coordinates of D, E, F, G and I are (0,0,3), (4,0,3), (4,12,4),