Dot product of vector and a dyadic.

If that is self-study, then surely the book(s) you use should explain the dot product before giving you examples of this kind? What exactly do you not understand in your books?

 

One vector is 4i+2j and the other is i+j right? Then those two vectors are not the same size. (And when I say 'size', I mean inner product).

 

When you multiply 2M on the left by a, you deal with the unit vectors on the left in each term of the dyad. Similarly, when you multiply on the right by b, you deal with the unit vectors on the right. That's all there is to it.

 

It is a rather strange book that you have. If it introduces something, it should explain how that something works. For a dyad [itex]\textbf{ab}[/itex] and a vector [itex]\textbf{c}[/itex]: [tex]\textbf{c} \cdot \textbf{ab} = (\textbf{c} \cdot \textbf{a})\textbf{b}[/tex][tex] \textbf{ab} \cdot \textbf{c}= \textbf{a}(\textbf{b} \cdot \textbf{c})[/tex]

 

Of course, 2M= 6ii + 4ij - 2ji + 4kj which can be represented as the matrix
[tex]\begin{bmatrix}6 & 4 & 0 \\ -2 & 0 & 0 \\ 0 & 4 & 0\end{bmatrix}[/tex]

Your product is the same as the matrix product
[tex]\begin{bmatrix}1 & -2 & 2\end{bmatrix}\begin{bmatrix}6 & 4 & 0 \\ -2 & 0 & 0 \\ 0 & 4 & 0\end{bmatrix}\begin{bmatrix}4\\ -3 \\ 0\end{bmatrix}[/tex]