[In this question, the unit vectors i and j are due east and due north respectively - Edexcel - A-Level Maths Mechanics - Question 7 - 2012 - Paper 1

The position vector p of boat P at time t is given by:

$p = (2i - 8j) + t(-4i + 8j) = (2 - 4t)i + (-8 + 8t)j$

To find when boat P is due west of boat Q, we equate the j-components of their position vectors:

For Q: $q = (18i + 12j) - (6i + 8j)$, thus: $q = (12i + 4j)$

For P when due west: The j-component is the same, set: $-8 + 8t = 12$ This leads to: 8t = 20 \\ t = rac{20}{8} = rac{5}{2} = 2.5 ext{ hours}

First, we calculate the position vectors of P and Q at t = 2.5:

For P: $p = (2 - 4(2.5))i + (-8 + 8(2.5))j = (-8)i + (12)j$

For Q: $q = (12i + 4j)$

Now, the distance between P and Q is given by the vector difference: Distance = $||p - q||$:

$p - q = [(-8) - 12]i + [12 - 4]j = (-20)i + (8)j$

Then, find the magnitude:

ormalsize egin{pmatrix} -20 \ 8 \\ ext{km} \end{pmatrix} = ormalsize rac{(-20^2 + 8^2)^{1/2}}{1} = ormalsize rac{(400 + 64)^{1/2}}{1} = rac{(464)^{1/2}}{1} ext{ km} \\ = 21.54 ext{ km approximately.}$$