21M.1.AHL.TZ1.13

pestleMathematicsAIHLPaper 121M· ahl-3-12-vector-applications-to-kinematicssource ↗

A submarine is located in a sea at coordinates $(0.8, 1.3, - 0.3)$ relative to a ship positioned at the origin $O$ . The $x$ direction is due east, the $y$ direction is due north and the $z$ direction is vertically upwards.

All distances are measured in kilometres.

The submarine travels with direction vector $(\begin{matrix} - 2 \\ - 3 \\ 1 \end{matrix})$ .

The submarine reaches the surface of the sea at the point $P$ .

Assuming the submarine travels in a straight line, write down an equation for the line along which it travels.

[2]

Find the coordinates of $P$ .

[3]

b.i.

Find $OP$ .

[2]

b.ii.

Markscheme / solution

$r = (\begin{matrix} 0.8 \\ 1.3 \\ - 0.3 \end{matrix}) + λ (\begin{matrix} - 2 \\ - 3 \\ 1 \end{matrix})$ A1A1

Note: Award A1 for each correct vector. Award A0A1 if their “ $r =$ ” is omitted.

[2 marks]

$- 0.3 + λ = 0$ (M1)

$\Rightarrow λ = 0.3$

$r = (\begin{matrix} 0.8 \\ 1.3 \\ - 0.3 \end{matrix}) + 0.3 (\begin{matrix} - 2 \\ - 3 \\ 1 \end{matrix}) = (\begin{matrix} 0.2 \\ 0.4 \\ 0 \end{matrix})$ (M1)

$P$ has coordinates $(0.2, 0.4, 0)$ A1

Note: Accept the coordinates of $P$ in vector form.

[3 marks]

b.i.

$\sqrt{0 . 2^{2} + 0 . 4^{2}}$ (M1)

$= 0.447 km (= 447 m)$ A1

[2 marks]

b.ii.

Examiners’ report

[N/A]

b.i.

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b.ii.