TITLE: What is meant by linearity of a dot product?
QUESTION [5 upvotes]: I would like to know what is meant by linearity of a dot product.
Thank you

REPLY [3 votes]: For a scalar $c$ and vectors $v,w,u$, 
$$
(c\, v+w)\cdot u=c(v\cdot u)+w\cdot u, \ \ u\cdot(c\,v+w)=c\,(u\cdot v)+u\cdot w.
$$
In other words, the dot product is linear in each of its two variables.

REPLY [3 votes]: It means that the dot product satisfies two properties:

If $u$, $v$ and $w$ are vectors such that $\cdot$ and $+$ make sense, $$u \cdot (v + w) = u \cdot v + u \cdot w$$ and vice-versa:
$$(u + v) \cdot w = u \cdot w + v \cdot w$$
If $u$ and $v$ are vectors such that $\cdot$ makes sense, and $c$ is a scalar, $$(cu) \cdot v = c (u \cdot v)$$ and vice-versa: $$u \cdot (cv) = c (u \cdot v)$$