- \hspace{0pt} [In this question, the horizontal unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are directed due east and due north respectively and position vectors are given relative to a fixed origin \(O\).]
Two speedboats, \(A\) and \(B\), are each moving with constant velocity.
- the velocity of \(A\) is \(40 \mathrm { kmh } ^ { - 1 }\) due east
- the velocity of \(B\) is \(20 \mathrm { kmh } ^ { - 1 }\) on a bearing of angle \(\alpha \left( 0 ^ { \circ } < \alpha < 90 ^ { \circ } \right)\), where \(\tan \alpha = \frac { 4 } { 3 }\) The boats are modelled as particles.
- Find, in terms of \(\mathbf { i }\) and \(\mathbf { j }\), the velocity of \(B\) in \(\mathrm { km } \mathrm { h } ^ { - 1 }\)
At noon
- the position vector of \(A\) is \(20 \mathbf { j } \mathrm {~km}\)
- the position vector of \(B\) is \(( 10 \mathbf { i } + 5 \mathbf { j } ) \mathrm { km }\)
At time \(t\) hours after noon
- the position vector of \(A\) is \(\mathbf { r k m }\), where \(\mathbf { r } = 20 \mathbf { j } + 40 t \mathbf { i }\)
- the position vector of \(B\) is \(\mathbf { s }\) km
- Find an expression for \(\mathbf { s }\) in terms of \(t , \mathbf { i }\) and \(\mathbf { j }\).
- Show that at time \(t\) hours after noon,
$$\overrightarrow { A B } = [ ( 10 - 24 t ) \mathbf { i } + ( 12 t - 15 ) \mathbf { j } ] \mathrm { km }$$
Show that the boats will never collide.Find the distance between the boats when the bearing of \(B\) from \(A\) is \(225 ^ { \circ }\)