We have two points $a,b \in \mathbb{R}^n$. The line connecting $ a$ and $b$ equals $$ \{ \lambda a + \mu b : \lambda, \mu \ \in \mathbb{R} \}$$ with $\lambda + \mu = 1$.
I want to prove this claim but I don't even understand why it is true. I'd expect the easiest way to find a vector through $a$ and $b$ to do $\vec{b} - \vec{a}$ and then shift the vector to pass through the points. I guess that you can change $\vec{a} + \vec{b}$ to $\vec{b} - \vec{a}$ using the coefficients $\mu$ and $\lambda$, but I also have no idea as to why these should add up to 1. Can anyone provide me with intuition regarding the above proposition?
Define $f:[0,1] \rightarrow \mathbb R^n$ by $f(t) = (\vec b - \vec a)t + \vec a$
This defines a line segment from $\vec a$ to $\vec b$ with $f(0) = \vec a$ and $f(1) = \vec b$.
We can rewrite this as $f(t) = \vec b t - \vec a t + \vec a = \vec b t + \vec a (1-t)$
Thus, $f(t) = (1-t)\vec a + t\vec b$
Let $\mu = t$ and $\lambda = 1-t$. Then $f(t) = \lambda \vec a + \mu \vec b$ and $\lambda + \mu = 1$.