Find a geometrical description of the set of all vectors of the form av+bw where a+b=1 My first intuition for solving this problem was to solve for $a$, therefore $a = 1-b$,
and letting $v$  and $w$ be arbitrary vertices $v =(1,3)$ and  $w = (5,2)$. Since the original question is $av+bw$ I just substitute $(1-b)*(1,3)+b(5,2) = ?$ Then multiply by using the dot product, $(1,-3b) + (5b+2b)$... I'm really stuck in here, now my question is am I doing this alright? and if so what do I do next? I also though of some how relating this to a span of vectors, which is $c_1v_1+c_2v_2+c_3v_3+\ldots+c_nv_n$ And if this is the case how do I start ? Thank you in advance (:

The set of points
$${\rm S}=\left\{t\vec{v}+\left(1-t\right)\vec{w}:t\in\mathbb{R}\right\}=\left\{\vec{w}+t\left(\vec{v}-\vec{w}\right):t\in\mathbb{R}\right\}$$
is a line characterized by the vector $\vec{v}-\vec{w}$ and the point $\vec{w}$.
