- 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 \(O\).
- Determine a vector equation of the plane that models the lawn, giving your answer in the form \(\mathbf { r } = \mathbf { a } + \lambda \mathbf { b } + \mu \mathbf { c }\)
- Show that, according to the model, the lawn is perpendicular to the vector \(\left( \begin{array} { c } 1
2
- 10 \end{array} \right)\) - Hence determine a Cartesian equation of the plane that models the lawn.
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 \(Q ( 6,4,3 )\).
- Determine a vector equation of the line that models the washing line.
- State a limitation of one of the models.
The point \(R ( 2,5,2.75 )\) lies on the washing line.
- Determine, according to the model, the shortest distance from the point \(R\) to the lawn, giving your answer to the nearest cm.
Given that the shortest distance from the point \(R\) to the lawn is actually 1.5 m ,
- use your answer to part (e) to evaluate the model, explaining your reasoning.