\documentclass[11pt]{article} % 

\usepackage{amsfonts,amstext}
\usepackage{amsthm}

\setlength{\oddsidemargin}{-0.15in}
\setlength{\topmargin}{-0.5in}
\setlength{\textwidth}{6.5in}
\setlength{\textheight}{9in}

\newcommand{\cP}{{\cal P}}
\newcommand{\IN}{{\bf N}}
\newcommand{\IZ}{{\bf Z}}
\newcommand{\IR}{{\bf R}}
\newcommand{\IQ}{{\bf Q}}
\newcommand{\points}[1]{{\it (#1 Points)}}


\begin{document} \openup .5\jot
%\maketitle

\noindent
{\Large Math 211 \hfill Homework 12 \quad Sample Texfile}

\medskip
\noindent{\bf 1.}
Compute $\left(\frac{{\bf x\cdot w}}{{\bf x\cdot x}}\right)
{\bf x}$, where
$${\bf x}=\left[\matrix{6\cr -2\cr 3}\right] \quad \hbox{ and } \quad 
{\bf w} =  
\left[\matrix{3\cr -1\cr -5}\right].$$

\medskip
\noindent{\bf 2.}
Find a unit vector in the direction of the vector 
${\bf x}=\left[\matrix{-6\cr 4\cr -3}\right]$.


\medskip
\noindent{\bf 3.} Let ${\bf u} = \left[\matrix{12\cr 3\cr -5}\right]$, ${\bf v} = 
\left[\matrix{2\cr -3\cr 3}\right]$, and 
${\bf 3} = \left[\matrix{3\cr -3\cr 1}\right]$.

Check which pairs of the above vectors are orthogonal.

\medskip
\noindent{\bf 4.}
Consider the set of vectors
$$\left\{ \left[\matrix{2\cr -5\cr -3}\right], 
\left[\matrix{4\cr -2\cr 6}\right], \left[\matrix{0\cr 0\cr 0}\right] 
\right\}.$$

(a) Show that it is  an orthogonal set.

(b) Find a nonzero vector orthogonal to the span of the set.



\medskip
\noindent{\bf 5.}
Show that  $\left\{ {\bf u_1,u_2,u_3}\right\}$ 
is an orthogonal basis for $\mathbb{R}^3$ if 
$${\bf u_1} =\left[\matrix{3\cr -3\cr 0}\right], \quad 
{\bf u_2} = \left[\matrix{2\cr 2\cr -1}\right], \quad
{\bf u_3}=\left[\matrix{1\cr 1\cr 4}\right].$$

Express
${\bf x}=\left[\matrix{5\cr -3\cr 1}\right]$ as a linear combination of 
${\bf u}_1$, ${\bf u}_2$, and ${\bf u}_3$.


\medskip
\noindent{\bf 6.}
Let  ${\bf y}=\left[\matrix{2\cr 6}\right]$ and ${\bf u}=
\left[\matrix{7\cr 1}\right]$. Express ${\bf y} = a{\bf u} + b{\bf z}$
such that ${\bf z}$ is orthogonal to ${\bf u}$.



\medskip
\noindent{\bf 7.} 
Let  ${\bf y} = \left[\matrix{-3\cr 9}\right]$ 
and ${\bf u}=
\left[\matrix{1\cr 2}\right]$.
Find the distance from ${\bf y}$ to the line passing through ${\bf u}$.



\medskip\noindent
{\bf 8.} Let  ${\bf x}_1 = \left[\matrix{-2/3\cr 1/3\cr 2/3\cr}\right]$ 
and ${\bf x}_2=
\left[\matrix{1/3\cr h \cr 0\cr}\right]$.

Determine $h$ so that the two vectors are orthogonal; 

then
find positive numbers $a, b$ so that 
$a{\bf x}_1$ and $b{\bf x}_2$ are orthonormal.


\end{document}
