- With respect to a fixed origin \(O\), the line \(l _ { 1 }\) is given by the equation
$$\mathbf { r } = \left( \begin{array} { r }
8
1
- 3
\end{array} \right) + \mu \left( \begin{array} { r }
- 5
4
3
\end{array} \right)$$
where \(\mu\) is a scalar parameter.
The point \(A\) lies on \(l _ { 1 }\) where \(\mu = 1\)
- Find the coordinates of \(A\).
The point \(P\) has position vector \(\left( \begin{array} { l } 1
5
2 \end{array} \right)\)
The line \(l _ { 2 }\) passes through the point \(P\) and is parallel to the line \(l _ { 1 }\) - Write down a vector equation for the line \(l _ { 2 }\)
- Find the exact value of the distance \(A P\).
Give your answer in the form \(k \sqrt { 2 }\), where \(k\) is a constant to be found.
The acute angle between \(A P\) and \(l _ { 2 }\) is \(\theta\)
- Find the value of \(\cos \theta\)
A point \(E\) lies on the line \(l _ { 2 }\)
Given that \(A P = P E\), - find the area of triangle \(A P E\),
- find the coordinates of the two possible positions of \(E\).