<html><head><meta name="color-scheme" content="light dark"></head><body><pre style="word-wrap: break-word; white-space: pre-wrap;">\documentclass[a4paper,12pt]{article}
\newcommand{\ds}{\displaystyle}
\newcommand{\pl}{\partial}
\parindent=0pt
\begin{document}


{\bf Question}

\begin{description}
\item[(a)] The position vectors of the points $P$ and $Q$ are
$${\bf a} + 3{\bf b} - {\bf c} {\rm \ \ and \ \ }3{\bf a} -{\bf
c}$$  Find $\vec{PQ}$ and $\vec{OM},$ where $M$ is the midpoint of
$PQ$, in terms of ${\bf a, b, c}$.
\item[(b)] For ${\bf d = a+b}$ and ${\bf e} = 3{\bf a + b - 4c}$
determine scalars $p$ and $q$ such that $$p{\bf d} + q{\bf e} = 10
{\bf a} + 10 {\bf b} - 12 {\bf c}$$ when ${\bf a, b, c}$ are
non-coplanar vectors.

\end{description}

\vspace{.25in}

{\bf Answer}

\begin{description}

\item[(a)]
${\bf p} = {\bf a} + 3{\bf b} - {\bf c} \hspace{.5in} {\bf q} =
3{\bf a} -{\bf c}$

$\vec{PQ} = {\bf q-p} = 2{\bf a} - 3{\bf b}$

$\vec{OM} = \frac{1}{2}({\bf p+q}) = 2{\bf a} + \frac{3}{2}{\bf b}
- {\bf c}$

${}$

\item[(b)]
\begin{eqnarray*} p{\bf d} + q{\bf e} &amp; = &amp; p{\bf a+b} + q(3{\bf a + b -
4c})\\ &amp; = &amp; (p+3q) {\bf a} + (p+q){\bf b} - 4q{\bf c} \\  &amp; = &amp; 
16{\bf a} + 10{\bf b} - 12{\bf c} \end{eqnarray*} So $q=3, \, p=7$
as ${\bf a, \, b, \, c},$ are independent.




\end{description}
\end{document}
</pre></body></html>