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\begin{document}
	
%	\mbox{}
%	\title{Geometry of Space and Vector Functions}
%	
%	\maketitle
	
	
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	%\section{Introduction} \label{sec:intro}
	%\noindent {\footnotesize Note: Most of the problems were taken from the textbook} \cite{jS15}.

\section*{Lines and Planes}
\bigskip

\begin{problem} [Stewart, Exercise~12.4.45] \label{prob:line from a point to a line}
	Let $P$ be a point not on the line $\ell$ that passes through the point $Q$ and $R$. Show that the distance $d$ from the point $P$ to the line $\ell$ is
	\[
		d = \frac{|a \times b|}{|a|},
	\]
	where $a$ and $b$ are the vectos $QR$ and $QP$, respectively. Use this formula to find the distance from $(1,1,1)$ to the line that passes through the points $(0,6,8)$ and $(-1,4,7)$.
\end{problem}

\begin{problem} [Stewart, Exercise~12.5.70]
	Find the distance from the point $(0,1,3)$ to the line
	\[
		\frac{x}{2} = \frac{6-y}{2} = z-3.
	\]
\end{problem}

\begin{problem} [Stewart, Exercises~12.5.(27,31, 37, 39)]
	Find the equation of the plane:
	\begin{enumerate}
		\item through the point $(2,0,1)$ and parallel to the plane $5x - y - z = 6$;
		\vspace{3pt}
		\item through the points $(0,1,1)$, $(1,0,1)$, and $(1,1,0)$;
		\vspace{3pt}
		\item that passes through the points $(3,1,4)$ and contains the line of intersection of the planes $x + 2y + 3z = 1$ and $2x - y + z = -3$;
		\vspace{3pt}
		\item that passes through the point $(1,5,1)$ and is perpendicular to the planes $2x + y - 2z = 2$ and $x + 3z = 4$.
	\end{enumerate}
\end{problem}

\begin{problem} [Stewart, Exercises~12.5.(41, 44)]
	Sketch the plane
	\begin{enumerate}
		\item $2x + 5y + z = 10$;
		\vspace{3pt}
		\item $3x + 2y = 12$;
		\vspace{3pt}
		\item $z = 7x$;
		\vspace{3pt}
		\item $6x + 5y - 3z = 15$.
	\end{enumerate}
\end{problem}

\begin{problem} [Stewart, Exercises~12.5.(75, 76)]
	Show that the distance between the parallel planes $ax + by + cz = d_1$ and $ax + by + cz = d_2$ is
	\[
		D = \frac{|d_1 - d_2|}{\sqrt{a^2 + b^2 + c^2}}.
	\]
	Find the equation of the planes that are parallel to the plane $x + 2y - 2z = 1$ and two units away from it.
\end{problem}

\begin{problem} [Stewart, Exercise~12.5.77]
	Show that the lines with symmetric equations $x=y=z$ and $x+1 = y/2 = z/3$ are skew. Find the distance between these lines.
\end{problem}
	
\vspace{20pt}

\begin{thebibliography}{20}
	
	\bibitem{jS15} J.~Stewart: \emph{Calculus} 8th Edition, Cengage Learning, Boston 2016.
	
\end{thebibliography}
	
\end{document}