Boat Vectors

slevvio · #1 $Old$ October 20th, 2008, 09:52 AM

A ship currently at position a is sailing on a straight course with velocity u. Simultaneously a submarine at position b manoeuvres to intercept the ship. If the ship maintains its straight course at all times, show that the intercept course is orthogonal to the direction (a - b) and that in order to the ship the submarine must have minimum speed

$\frac{|\textbf{u} \times (\textbf{a} - \textbf{b})|}{|\textbf{a} - \textbf{b}|}$ .

I'm not too worried about this as I have the answers right in front of me; however the bit I am having trouble with is this part:

$\textbf{d} =$ intercept course $= \textbf{x}_{A}(t) - \textbf{b} = (\textbf{a} - \textbf{b}) - \frac{|\textbf{a}-\textbf{b}|^2}{\textbf{u} \cdot (\textbf{a} - \textbf{b})}\textbf{u} = -\frac{(\textbf{a} - \textbf{b}) \times (\textbf{u} \times (\textbf{a} - \textbf{b}))}{|\textbf{a}-\textbf{b}|^2}$

I don't understand how $(\textbf{a} - \textbf{b}) - \frac{|\textbf{a}-\textbf{b}|^2}{\textbf{u} \cdot (\textbf{a} - \textbf{b})}\textbf{u} = -\frac{(\textbf{a} - \textbf{b}) \times (\textbf{u} \times (\textbf{a} - \textbf{b}))}{|\textbf{a}-\textbf{b}|^2}$ and any help would be appreciated.