I'm reading a book about tensor calculus (Tesor Calculus for Physicists, Dwight E. Neuenschwander) that introduces dyads as a short side topic in one of the exercises (2.11). I am confused about how they are defining how a dyad can act on a vector through the regular dot product. They say $C\cdot AB$ means $(C\cdot A)B$ (where $AB$ is the dyad — I see this might be also called a dyadic, but my book says dyad).
I'm given the vectors
\begin{equation} A=3i+2j+5k\\ B=7i+11k\\ C=2i+3j+4k \end{equation}
and being asked to calculate $AB\cdot C$, which I assume means $A(B\cdot C)$. What I want to do when I see this is
\begin{equation} A(B\cdot C)=A(14+44)=58A \end{equation}
However, I know this is incorrect because my textbook states that the first term should be $42i$. This seems to be what you get if you take the first term of $AB$, which is $21ii$, and do $21i(i\cdot C)$. I assume you then do the same for every term of $AB$. But then, why make the assertion that $AB\cdot C = A(B\cdot C)$? That seems to mean something completely different. Any help?
Your textbook is wrong. From the equation $$\begin{align} AB\cdot C &= (3i+2j+5k)(7i+11k)\cdot(2i+3j+4k) \\ &= (21ii+33ik+14ji+22jk+35ki+55kk)\cdot(2i+3j+4k) \end{align}$$ and the condition that this last product should be expanded by distributivity, you can see that he is failing to consider the $ji$, $ki$ and $ik$ terms of $AB$, all of which will contribute to the $i$ component of $AB\cdot C$.
You can read more on the relevant wiki article.