A boat \(B\) is moving with constant velocity. At noon, \(B\) is at the point with position vector \(( 3 \mathbf { i } - 4 \mathbf { j } ) \mathrm { km }\) with respect to a fixed origin \(O\). At 1430 on the same day, \(B\) is at the point with position vector \(( 8 \mathbf { i } + 11 \mathbf { j } ) \mathrm { km }\).
Find the velocity of \(B\), giving your answer in the form \(p \mathbf { i } + q \mathbf { j }\).
At time \(t\) hours after noon, the position vector of \(B\) is \(\mathbf { b } \mathrm { km }\).
Find, in terms of \(t\), an expression for \(\mathbf { b }\).
Another boat \(C\) is also moving with constant velocity. The position vector of \(C\), \(\mathbf { c k m }\), at time \(t\) hours after noon, is given by
$$\mathbf { c } = ( - 9 \mathbf { i } + 20 \mathbf { j } ) + t ( 6 \mathbf { i } + \lambda \mathbf { j } ) ,$$
where \(\lambda\) is a constant. Given that \(C\) intercepts \(B\),
find the value of \(\lambda\),
show that, before \(C\) intercepts \(B\), the boats are moving with the same speed.