7 Turn over Question 3 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 3 is 5 marks) 8 4. All units in this question are in metres. A lawn is modelled as a plane that contains the points L (–2, –3, –1), M (6, –2, 0) and N (2, 0, 0), relative to a fixed origin O. (a) Determine a vector equation of the plane that models the lawn, giving your answer in the form r = a + λb + μc (3) (b) (i) Show that, according to the model, the lawn is perpendicular to the vector 1 2 10 −         (ii) Hence determine a Cartesian equation of the plane that models the lawn. (4) There are two posts set in the lawn. There is a washing line between the two posts. The washing line is modelled as a straight line through points at the top of each post with coordinates P (–10, 8, 2) and Q (6, 4, 3). (c) Determine a vector equation of the line that models the washing line. (2) (d) State a limitation of one of the models. (1) The point R (2, 5, 2.75) lies on the washing line. (e) Determine, according to the model, the shortest distance from the point R to the lawn, giving your answer to the nearest cm. (2) Given that the shortest distance from the point R to the lawn is actually 1.5 m, (f) use your answer to part (e) to evaluate the model, explaining your reasoning. 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