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 : 5