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\begin{document}

{\bf Question}

Which of the following pairs of vector are parallel and which are
anti-parallel?
\begin{description}
\item[(i)]
${\bf{a}}={\bf{i}}-3{\bf{j}}+{\bf{k}}\
{\bf{b}}=-4{\bf{i}}+12{\bf{j}}-4{\bf{k}}$

\item[(ii)]
${\bf{a}}=-2{\bf{i}}+3{\bf{j}}-{\bf{k}}\
{\bf{b}}=2{\bf{i}}-3{\bf{j}}+{\bf{k}}$

\item[(iii)]
${\bf{a}}=4{\bf{i}}-{\bf{j}}-3{\bf{k}}\
{\bf{b}}=8{\bf{i}}-2{\bf{j}}-6{\bf{k}}$

\item[(iv)]
${\bf{a}}={\bf{i}}+7{\bf{j}}+{\bf{k}}\
{\bf{b}}=3{\bf{i}}+21{\bf{j}}+3{\bf{k}}$
\end{description}

\medskip

{\bf Answer}

Can be written as $\lambda{\bf{a}}$ where $\lambda$ is a scalar.
If so, then if $\lambda&gt;0$ the vectors are parallel. If so but
$\lambda&lt;0$ then they're anti-parallel. If \un{not} then they're
not parallel.

\begin{description}
\item[(i)]
${\bf{a}}={\bf{i}}-3{\bf{j}}+{\bf{k}},\
{\bf{b}}=-4{\bf{i}}+12{\bf{j}}-4{\bf{k}}=-4({\bf{j}}-3{\bf{k}}+{\bf{j}})
\Rightarrow \lambda=-4 \Rightarrow$ anti-parallel.

\item[(ii)]
${\bf{a}}=-(2{\bf{i}}-3{\bf{j}}+{\bf{k}})=-{\bf{b}} \Rightarrow$
anti-parallel.

\item[(iii)]
${\bf{b}}=8{\bf{i}}-2{\bf{j}}-6{\bf{k}}=2(4{\bf{i}}-{\bf{j}}-3{\bf{k}})=2{\bf{a}}
\Rightarrow$ parallel

\item[(iv)]
${\bf{a}}={\bf{i}}+7{\bf{j}}+{\bf{k}},\
{\bf{b}}=3{\bf{i}}+21{\bf{j}}+3{\bf{k}}=3({\bf{j}}+7{\bf{j}}+{\bf{k}})=3{\bf{a}}
\Rightarrow$ parallel

\end{description}
\end{document}
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