\documentclass[12pt]{article}
\usepackage{fullpage,amsmath,amsfonts,graphicx}

\def\d{{\mathrm d}}
\def\e{{\mathrm e}}
\def\R{{\mathbb R}}
\def\ba{\boldsymbol{a}}
\def\bb{\boldsymbol{b}}
\def\bc{\boldsymbol{c}}
\def\bd{\boldsymbol{d}}
\def\bn{\boldsymbol{n}}
\def\bu{\boldsymbol{u}}
\def\bv{\boldsymbol{v}}
\def\bw{\boldsymbol{w}}
\def\bp{\boldsymbol{p}}
\def\bx{\boldsymbol{x}}

\begin{document}

\title{Calculus and Elements of Linear Algebra I}
\author{Homework 10}
\date{Due on Moodle, Monday, November 23, 2020}
\maketitle


\begin{enumerate}

\item Prove the following identities for vectors $\ba, \bb, \bc
\in \R^3$. 
\begin{enumerate}
\item The
\emph{``BAC--CAB-identity''}
\[
  \ba \times (\bb \times \bc)
  = \bb \, (\ba \cdot \bc) - \bc \, (\ba \cdot \bb) \,.
\]
\item The \emph{Jacobi identity} in three dimensions
\[
  \ba \times (\bb \times \bc)
  + \bb \times (\bc \times \ba)
  + \bc \times (\ba \times \bb) = 0 \,.
\]
\end{enumerate}


\item Prove the following identities for vectors $\ba, \bb, \bc, \bd
\in \R^3$. 
\begin{enumerate}
  
\item The \emph{Cauchy--Binet formula} in three dimensions
\[
  (\ba \times \bb) \cdot (\bc \times \bd)
  = (\ba \cdot \bc) \, (\bb \cdot \bd)
  - (\ba \cdot \bd) \, (\bb \cdot \bc) \,.
\]

\emph{Hint:} Use the identity $\bu \cdot (\bv \times \bw) = \bv \cdot
(\bw \times \bu)$.

\item The identity
\[
  \lVert \ba \times \bb \rVert^2
  = \lVert \ba \rVert^2 \, \lVert \bb \rVert^2
  - (\ba \cdot \bb)^2 \,.
\]
\end{enumerate}


\item Find an equation for the plane that contains the point
$\bp=(2,4,6)$ and the line
\[
  \bx =
  \begin{pmatrix}
  7 \\ 3 \\ 5
  \end{pmatrix}
  + \lambda \,
  \begin{pmatrix}
  -3 \\ 4 \\ 2
  \end{pmatrix} \,.
\]

\item Find the distance between the point $\bp=(1,2,3)$ and the
line 
\[
  \bx =
  \begin{pmatrix}
  -1 \\ 1 \\ 6
  \end{pmatrix}
  + \lambda \,
  \begin{pmatrix}
  1 \\ -1 \\ 0
  \end{pmatrix} \,.
\]


\item Prove the following statement: Let $\bv_1, \dots, \bv_n$ be
linearly independent.  If a vector $\bw$ can be written 
\[
  \bw = \sum_{k=1}^n \alpha_k \, \bv_k \,,
\]
the choice of the coefficients $\alpha_1,\dots,\alpha_n$ is unique.


\end{enumerate}

\end{document}
