- \hspace{0pt} [In this question \(\mathbf { i }\) and \(\mathbf { j }\) are horizontal unit vectors directed due east and due north respectively and position vectors are given relative to a fixed origin \(O\).]
At midnight, a ship \(S\) is at the point with position vector ( \(19 \mathbf { i } + 22 \mathbf { j }\) )km
The ship travels with constant velocity \(( 12 \mathbf { i } - 16 \mathbf { j } ) \mathrm { km } \mathrm { h } ^ { - 1 }\)
- Find the speed of \(S\).
At time \(t\) hours after midnight, the position vector of \(S\) is \(\mathbf { s } \mathrm { km }\).
- Find an expression for \(\mathbf { s }\) in terms of \(\mathbf { i } , \mathbf { j }\) and \(t\).
A lighthouse stands on a small rocky island. The lighthouse is modelled as being at the point with position vector \(( 26 \mathbf { i } + 15 \mathbf { j } ) \mathrm { km }\).
It is not safe for ships to be within 1.3 km of the lighthouse.
- Find the value of \(t\) when \(S\) is closest to the lighthouse.
- Hence determine whether it is safe for \(S\) to continue its course.