23 Vectors 3 APQ Soln (RVHS)

13 Additional Practice Questions 1. (J92/I/6) The lines 1l and 2l have equations 3 1 1 2 , 0 4 t t                   r and 1 2 1 1 , 1 1 s s                    r  Show that 1l passes through the point 2, 1, 4  but that 2l does not pass through this point. Find the acute angle between 2l and the line joining the points 1, 1, 1 and 2, 1, 4  , giving your answers correct to the nearest degree. [Solution] Since 2 3 1 1 1 2 1 4 0 4 t t                               , 1l passes through the point 2, 1, 4  . Since we cannot find a value s such that 2 1 2 1 1 1 4 1 1 s                               , thus 2l does not pass through the point 2, 1, 4  . Acute angle between 2l and the line joining the points 1, 1, 1 and 2, 1, 4  1 1 2 11 1cos 1 06 261 5 7cos 55.9 566 26                          2. (N89/I/16) The points A and B have position vectors 4 2 4 i j k and 3 6 6 i j k respectively, relative to the origin O. The point P is such that 1 2OP OA    and the point Q is such that 1 3OQ OB   . The point C is such that OACB is a parallelogram. (i) Find the length AB and the cosine of the angle between OA  and OB . (ii) Find a vector equation, in parametric form, of the line PQ. (iii) Find the position vector of the point of intersection of PQ and OC. [Solution]

14 4 2 4 OA         , 3 6 6 OB         (i) 1 4 2 AB         . Thus 21AB  . cos 48 8 936 81 OA OBAOB OA OB             (ii) 4 21 2 12 4 2 OP                    , 3 11 6 23 6 2 OQ                    1 1 0 PQ         Vector equation of line PQ is 2 1 1 1 , 2 0                     r  (iii) 7 8 10 OC OA OB                Vector equation of line OC is 7 8 , 10          r  2 1 7 1 1 8 2 0 10                              2 7 3 11 8 , 5 52 10                 position vector of the point of intersection is 71 85 10       .

15 . Relative to an origin O, points A and B have position vectors 2i j and 3 bj k respectively, where b > 0. Given that the angle between a and b is 120, find the exact value of b. For the following questions, take b = 2. Write down a vector equation of the line passing through A and B. (i) Find the position vector of the point P on the line AB such that OP  is perpendicular to .AB  (ii) A point Q is on AB produced. Find the position vector of Q such that AB BQ . Solution: 1 2 0 OA         , 0 3OB b         2 2 2 2 2 1 0 2 3 1 4 0 0 9