- Two birds are flying towards their nest, which is in a tree.
Relative to a fixed origin, the flight path of each bird is modelled by a straight line.
In the model, the equation for the flight path of the first bird is
$$\mathbf { r } _ { 1 } = \left( \begin{array} { r }
- 1 \\
5 \\
2
\end{array} \right) + \lambda \left( \begin{array} { l }
2 \\
a \\
0
\end{array} \right)$$
and the equation for the flight path of the second bird is
$$\mathbf { r } _ { 2 } = \left( \begin{array} { r }
4 \\
- 1 \\
3
\end{array} \right) + \mu \left( \begin{array} { r }
0 \\
1 \\
- 1
\end{array} \right)$$
where \(\lambda\) and \(\mu\) are scalar parameters and \(a\) is a constant.
In the model, the angle between the birds’ flight paths is \(120 ^ { \circ }\)
- Determine the value of \(a\).
- Verify that, according to the model, there is a common point on the flight paths of the two birds and find the coordinates of this common point.
The position of the nest is modelled as being at this common point.
The tree containing the nest is in a park.
The ground level of the park is modelled by the plane with equation
$$2 x - 3 y + z = 2$$ - Hence determine the shortest distance from the nest to the ground level of the park.
- By considering the model, comment on whether your answer to part (c) is reliable, giving a reason for your answer.