Write down a vector equation for L in the form ${\boldsymbol{r}} = {\boldsymbol{a}} + t{\boldsymbol{b}}$ .

Find

(i)     $\overrightarrow {{\rm{OP}}}$ ;

(ii)    $|\overrightarrow {{\rm{OP}}} |$ .

correct equation in the form ${\boldsymbol{r}} = {\boldsymbol{a}} + t{\boldsymbol{b}}$     A2     N2

${\boldsymbol{r}} = \left( {\begin{array}{*{20}{c}} 1\\ { - 1}\\ 2 \end{array}} \right) + t\left( {\begin{array}{*{20}{c}} 1\\ 3\\ { - 2} \end{array}} \right)$

[2 marks]

(i) attempt to substitute $t = 2$ into the equation     (M1)

e.g. $\left( {\begin{array}{*{20}{c}} 2\\ 6\\ { - 4} \end{array}} \right)$ , $\left( {\begin{array}{*{20}{c}} 1\\ { - 1}\\ 2 \end{array}} \right) + 2\left( {\begin{array}{*{20}{c}} 1\\ 3\\ { - 2} \end{array}} \right)$

$\overrightarrow {{\rm{OP}}}  = \left( {\begin{array}{*{20}{c}} 3\\ 5\\ { - 2} \end{array}} \right)$     A1     N2

(ii) correct substitution into formula for magnitude     A1

e.g. $\sqrt {{3^2} + {5^2} + - {2^2}}$ , $\sqrt {{3^2} + {5^2} + {2^2}}$

$|\overrightarrow {{\rm{OP}}}|  = \sqrt {38}$     A1     N1

[4 marks]

Many candidates answered this question well. Some continue to write the vector equation in (a) using "L =", which does not earn full marks.

Part (b) proved accessible for most, although small arithmetic errors were not uncommon. Some candidates substituted $t = 2$ into the original equation, and a few answered  $\overrightarrow {{\rm{OP}}} = \left( {\begin{array}{*{20}{c}} 2\\ 6\\ { - 4} \end{array}} \right)$ . A small but surprising number of candidates left this question blank, suggesting the topic was not given adequate attention in course preparation.