RI Vectors Lines and Planes 7B Solns

Page 1 RAFFLES INSTITUTION H2 Mathematics 9758 2023 Year 6 Term 3 Revision 7b (Tutorial) Topic: Vectors 3 (Lines and Planes) Summary for Lines and Planes [Refer to Revision 7a] Revision Tutorial Questions Source of Question: NJC Prelim 9758/2018/02/Q2 1 The planes 1p and 2 ,p have equations 2360xyz++= and 1 26 2   = r  respectively. (i) Find a vector equation of the line of intersection, l , between 1p and 2.p [2] The line m passes through the points ( )2 ,1 ,1A and ( )5 ,4 ,2 .B (ii) Verify that A lies on 2.p [1] (iii) Find the coordinates of the points on m that are equidistant from planes 1p and 2p . [5] Solution (i) 1 : 2 3 6 0p xyz ++= 2 : 226px y z++= Using GC, 18 6 12 2 , 01 λλ −−     = +∈       r  (ii) Since 2 2(1) 2(1) 6,++= or 21 1 2 2226 , 12   =++=  the point A lies on 2.p (iii) Let the point that is equidistant from both planes be C. 523 41 3 211  −=

Page 2 23 13 11 OC t    = +      for some t∈  Distance of C from 1p = Distance of C from 2p 222 222 23 2 1 23 0 2 13 1 2 13 0 3 11 2 10 6 122 236 3 6 2 46 39 66 37 11 13 21 37 77 39 63 tt tt tt ttt t t t tt tt +  +          +− ⋅ +− ⋅          ++      = ++ ++ ++ ++ ++ += += = + 77 39 63 or 77 39 63 140 39 or 14 39 39 39 or 140 14 t tt t tt tt = −− =+ = −= =−= 2 3 163 39 11 3 23140 1401 1 101 OC        = +− =            or 2 3 145 39 11 3 13114 141 1 53 OC        = +=             The two points are 163 23 101,,140 140 140   and 145 131 53,,14 14 14   . Source of Question: NJC JC2 Mid-Year CT 9758/2018/01/Q6 2 The equation of the plane p is given by 2 5 7,ax y z−−= where a is a real constant. (i) Given that the line l with equation 35 5 2 , 60 λλ      = +− ∈       r intersects the plane p exactly once, find the possible values of a. [1] Assume for the remainder of this question that a = 3. (ii) Find the coordinates of the point of intersection between l and p. [3] (iii) Find the acute angle between the line l and the plane p. [2] (iv) Find the vector equation of the line of reflection of the line l in the plane p. [5]

Page 3 Solution (i) Since 5 42 2 0 5 40 505 a aa    −⋅ − ≠⇒ +≠⇒≠ −   −  a can be any real number except 4 5− . (ii) Let N be the point of intersection between l and p. Since N lies on l, 35 52 6 λ λ + = −  ON for some .λ∈  Since N also lies on plane p, 3 27 5  ⋅− =−  ON 35 3 52 . 2 7 65 9 15 10 4 30 7 19 7 9 10 30 38 2 λ λ λλ λ λ +     − −=     −   + −+−= =−+ + = = Hence, 3 5(2) 13 5 2(2) 1 . 66 +   = −=    