A boat $B\,$ is moving with constant velocity. At noon, $B\,$ is at the point with position vector $(3i−4j)(3\mathbf{i} - 4\mathbf{j})$ km with respect to a fixed origin $O$ . At 1430 on the same day, $B\,$ is at the point with position vector $(8i+11j)(8\mathbf{i} + 11\mathbf{j})$ km.

Find the velocity of $B$ , giving your answer in the form $pi+qjp\mathbf{i} + q\mathbf{j}$ .

At time $t\,$ hours after noon, the position vector of $B\,$ is $b\mathbf{b}$ km. Find, in terms of $t$ , an expression for $b\mathbf{b}$ .

Another boat $C\,$ is also moving with constant velocity. The position vector of $C$ , $c\mathbf{c}$ km, at time $t\,$ hours after noon, is given by $c=(−9i+20j)+t(6i+λj)\mathbf{c} = (-9\mathbf{i} + 20\mathbf{j}) + t(6\mathbf{i} + \lambda\mathbf{j})$ , where $\lambda\,$ is a constant. Given that $C\,$ intercepts $B$ , find the value of $λ\lambda$ .

Show that, before $C\,$ intercepts $B$ , the boats are moving with the same speed.

Find the velocity of $B$ , giving your answer in the form $pi+qjp\mathbf{i} + q\mathbf{j}$ .

At time $t\,$ hours after noon, the position vector of $B\,$ is $b\mathbf{b}$ km. Find, in terms of $t$ , an expression for $b\mathbf{b}$ .

Show that, before $C\,$ intercepts $B$ , the boats are moving with the same speed.

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