2. 2022 Vectors (Lines) Discussion Solution

CJC MATHEMATICS DEPARTMENT 2022 JC1 H2 MATHEMATICS (9758) TOPIC: VECTORS Page | 1 H2 MATHEMATICS (9758) TOPIC VECTORS (LINES) 2022/JC1 DISCUSSION 1 Find a vector equation and cartesian equation of the following lines: (a) passing through the point with position vector 7i + 2j – 4k and parallel to to i – 3j + k (b) passing through the points (2, –2, 1) and (0, 4, 9 ) (c) passing through the point with position vector 7 i and parallel to the line r = (2 – p) i + pj + 5k. Solution: (a) Vector equation of line: 71 r 2 3 , 41          = + −      −    Cartesian equation of line: 7 2 ( 4) 1 3 1 274 3 − − − −==− −− = = +− x y z yxz (b) 2 0 2 1 Direction of line = 2 4 6 2 3 1 9 8 4 −               − − = − =−              −        Equation of line: 21 r 2 3 , 14   −       = − +          Cartesian equation of line: 2 ( 2) 1 1 3 4 212 34 − − − −==− +−− = = x y z yzx (c) Equation of parallel line: 2 2 2 1 r 0 0 1 5 5 0 5 0 − − −                   = = + = +                            pp p p p 1 Direction of line = 1 0 −   Equation of line: 71 r 0 1 , 00   −       =+          Cartesian equation of line: 70 , 011 7 , 0 −− ==− − = = xy z x y z

CJC MATHEMATICS DEPARTMENT 2022 JC1 H2 MATHEMATICS (9758) TOPIC: VECTORS Page | 2 2 Convert the equations of following lines to their vector equation form: (a) 13 2 6 2 −= − = zyx (b) 5 2 4 , 3+ = − =x y z Solution: (a) 13 2 6 2 −= − = zyx 2 6 3 1 1 1 2 3 0 1 11 2 23 −− == − − −== x y z x y z Vector equation of line: 1 2 '' 1 3 3 r 0 , 12   =+     33 r 0 2 , 1 12          =+          (b) 5 2 4 , 3+ = − =x y z ( 5) 4 2 , 311 1 ( 5) 2 , 311 4 − − −== − −−− == − xy z yx z Vector equation of line: ''11 24 51 r , 30   −       = + −          1 2 54 r 1 , 30   −       = + −         

CJC MATHEMATICS DEPARTMENT 2022 JC1 H2 MATHEMATICS (9758) TOPIC: VECTORS Page | 3 3 Given three points ( )0,2,7A , ( )5, 3, 2B − and ( )1, 1, 1C , find the position vector of the point R on AB such that CR is perpendicular to AB . Hence find the perpendicular distance from C to AB and the position vector of the reflection of C in AB . Solution: 5 0 5 1 3 2 5 5 1 2 7 5 1 AB −               = − − = − =−              −        Equation of line AB : 01 r 2 1 , 71  −       =+      