- #1

- 996

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## Homework Statement

Using index notation only (no expanding of terms), show:

\begin{equation*}

\text{(a) }\underline{ \bf{a}} \cdot \underline{\bf{A}} \underline{\bf{b}} = \underline{\bf{A}} \cdot \underline{\bf{a}} \otimes \underline{\bf{b}}

\end{equation*}

## Homework Equations

\begin{align*}

& \underline{\bf{A}} = A_{ij}(\underline{\bf{e}}_i \otimes \underline{\bf{e}}_j) \\

& \underline{\bf{a}}\otimes \underline{\bf{b}} = a_i b_j (\underline{\bf{e}}_i \otimes \underline{\bf{e}}_j)

\end{align*}

## The Attempt at a Solution

I'm actually quite close but ran into trouble:

\begin{align*}

& \text{(1) } \underline{\bf{A}} \underline{\bf{b}} =

A_{ij}(\underline{\bf{e}}_i \otimes \underline{\bf{e}}_j) b_k \underline{\bf{e}}_k \\

&\text{(2) }\underline{ \bf{a}} \cdot \underline{\bf{A}} \underline{\bf{b}} =

A_{ij}(\underline{\bf{e}}_i \otimes \underline{\bf{e}}_j) b_k \underline{\bf{e}}_k a_p \underline{\bf{e}}_p \\

&\text{(3) }\underline{ \bf{a}} \cdot \underline{\bf{A}} \underline{\bf{b}} =

A_{ij}(\underline{\bf{e}}_i \otimes \underline{\bf{e}}_j)( a_p b_k \underline{\bf{e}}_k \underline{\bf{e}}_p)

\end{align*}

The right term in (3) looks very close to [itex]\underline{\bf{a}}\otimes \underline{\bf{b}}[/itex] but not quite. What am I doing wrong?

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