In the Rotations section of Arfken's chapter on vector analysis in the book "Mathematical Methods for Physicists", there is a statement which says the following -
The dot product is the projection of $\bf{e_{u'}}$ onto the $\bf{e_v}$ direction and is therefore the change in $x_v$ that is produced by a unit change in $x_{u'}$.
Although it sounds very trivial, and it probably is, I am unable to understand this statement clearly. Can someone explain why the statement is true?
For reference, this statement is on page 139 in the 7th edition of the book.
Edit: $e_{u^\prime}\cdot e_v$ is the dot product involved
Let $e_{v_1}, ... e_{v_n}$ be an orthogonal basis, and $x_{v_i}$ the coordinate functions so that any vector $w$ decomposes as $$w = \sum_{i=1}^n x_{v_i}(w) e_{v_i}.$$ Take a dot product of both sides with a basis vector $e_{v_i}$ and use orthogonality to see that $$x_{v_i}(w) = w \cdot e_{v_i}. \tag{$*$} $$
The quote you gave is articulating $$x_{v_i}(w + e_{u'}) = x_{v_i}(w) + e_{u'} \cdot e_{v_i},$$ which follows immediately from $(*)$ above.