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.