Show that the two submarines would collide at a point P and write down the coordinates of P.

Show that submarine B travels in the same direction as originally planned.

Find the value of t when submarine B passes through P.

Find an expression for the distance between the two submarines in terms of t.

Find the value of t when the two submarines are closest together.

Find the distance between the two submarines at this time.

r_A = r_{B        (M1)}

2 − t = − 0.5t ⇒ t = 4       A1

checking t = 4 satisfies 4 + t = 3.2 + 1.2t and − 1 − 0.15t = − 2 + 0.1t      R1

P(−2, 8, −1.6)      A1

Note: Do not award final A1 if answer given as column vector.

[4 marks]

$0.9 \times \left( \begin{gathered} - 0.5 \hfill \\ \,1.2 \hfill \\ \,0.1 \hfill \\ \end{gathered} \right) = \left( \begin{gathered} - 0.45 \hfill \\ \,1.08 \hfill \\ \,0.09 \hfill \\ \end{gathered} \right)$     A1

Note: Accept use of cross product equalling zero.

hence in the same direction      AG

[1 mark]

$\left( \begin{gathered} \, - 0.45t \hfill \\ 3.2 + 1.08t \hfill \\ - 2 + 0.09t \hfill \\ \end{gathered} \right) = \left( \begin{gathered} - 2 \hfill \\ \,8 \hfill \\ - 1.6 \hfill \\ \end{gathered} \right)$      M1

Note: The M1 can be awarded for any one of the resultant equations.

$\Rightarrow t = \frac{{40}}{9} = 4.44 \ldots$     A1

[2 marks]

r_A − r_B = $\left( \begin{gathered} \,2 - t \hfill \\ \,4 + t \hfill \\ - 1 - 0.15t \hfill \\ \end{gathered} \right) - \left( \begin{gathered} \, - 0.45t \hfill \\ 3.2 + 1.08t \hfill \\ - 2 + 0.09t \hfill \\ \end{gathered} \right)$      (M1)(A1)

$= \left( \begin{gathered} \,2 - 0.55t \hfill \\ \,0.8 - 0.08t \hfill \\ 1 - 0.24t \hfill \\ \end{gathered} \right)$     (A1)

Note: Accept r_A − r_B.

distance $D = \sqrt {{{\left( {2 - 0.55t} \right)}^2} + {{\left( {0.8 - 0.08t} \right)}^2} + {{\left( {1 - 0.24t} \right)}^2}}$      M1A1

$\left( { = \sqrt {8.64 - 2.688t + 0.317{t^2}} } \right)$

[5 marks]

minimum when $\frac{{{\text{d}}D}}{{{\text{d}}t}} = 0$      (M1)

t = 3.83      A1

[2 marks]

0.511 (km)      A1

[1 mark]