Skip to main content Contents Index Prev Up Next \(\newcommand{\val}{Val\text{00E9}ry}
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Problems A.2 Problems
1.
Prove Jacobi’s identity: given three arbitrary vectors in an oriented vector space
\begin{equation*}
(\bU \times \bV) \times \bW + (\bV \times \bW) \times \bU + (\bW \times \bU)
\times \bV = \bze
\end{equation*}
2.
Given four vectors \(\bA\text{,}\) \(\bB\text{,}\) \(\bC\) and \(\bD\text{,}\) prove the identity
\begin{equation*}
(\bA\times\bB)\cdot(\bC\times\bD) = (\bA\times\bC)\cdot(\bB\times\bD) +(\bA\cdot\bC)(\bB\cdot\bD)- (\bA\cdot\bB)(\bC\cdot\bD)
\end{equation*}
3.
Find the conditions satisfied by three non-zero vectors \(\bU\text{,}\) \(\bV\) and \(\bW\) to guarantee the following equality
\begin{equation*}
(\bU \times \bV) \times \bW = \bU \times (\bV \times \bW)
\end{equation*}
4.
Given four arbitrary points \(P\text{,}\) \(Q\text{,}\) \(R\) and \(S\) prove the identity
\begin{equation*}
\br_{QR}\times \br_{QS} = \br_{PQ}\times \br_{PR} +\br_{PR}\times \br_{PS}
+ \br_{PS}\times \br_{PQ}
\end{equation*}
where \(\br_{AB}\) is the vector which points from \(A\) to \(B\text{.}\)
5.
In a Euclidean vector space \(E\text{,}\) a linear operator \(\cL: \bU \mapsto \cL(\bU)\) is said to be skew-symmetric if and only if \(\cL(\bU)\cdot \bV = - \bU \cdot \cL(\bV)\text{.}\)
Show that any skew-symmetric operator
\(\cL\) is represented by a skew-symmetric matrix relative to any orthonormal basis, that is, satisfying
\begin{equation*}
[\cL]_b^T =- [\cL]_b
\end{equation*}
Conversely, show that to a skew-symmetric matrix corresponds a skew-symmetric linear operator.
Show that, given a non-zero vector \(\bU\text{,}\) the linear operator \(\bX \mapsto \bU\times \bX\) is skew-symmetric. Conversely, show that all skew-symmetric operator are of the form \(\bX \mapsto \bU\times \bX\text{.}\)
6.
Given two non-zero vectors \(\bA\) and \(\bB\text{,}\) find the set of vectors \(\bX\) solution of the equation
\begin{equation*}
\bA \times \bX = \bB
\end{equation*}
7.
Given a non-zero vector \(\bY\text{,}\) consider the linear operator \(\cL_\bY\) defined by
\begin{equation*}
\cL_\bY (\bX) = \bX \times \bY
\end{equation*}
Find the matrix of \(\cL_\bY\) on right-handed orthonormal basis \(b (\be_1 , \be_2 , \be_3)\text{.}\)
Show that \(\cL_\bY^3 = - \bY^2 \cL_\bY\text{.}\)
Deduce an expression of operator \(\exp (\cL_\bY)= \cI +\cL_\bY+ \frac{1}{2!}\cL_\bY^2 +\frac{1}{3!}\cL_\bY^3 +\cdots\text{.}\)
