On page 49 of Gravitation by Misner, Thorne and Wheeler they talk about the different ways of representing a vector in spacetime. In Box C they give the equation for a parametrised straight line between points A and B as $$P\left(\lambda\right)=A+\lambda\left(B-A\right)$$ At my high-school-type maths level, I don't recognise this equation. For example, for the straight line $$y=3x+7$$ I would write $x=t$ and $y=3t+7$. So how to understand $P\left(\lambda\right)=A+\lambda\left(B-A\right)$? Thanks.
$P(\lambda)$ is parametrised by $\lambda$, where $\lambda=0$ corresponds to $A$ and $\lambda=1$ to $B$. The points for other $\lambda$ values are defined by linear interpolation between $A$ and $B$ – for example $\lambda=\frac34$ corresponds to a point $\frac34$ of the way from $A$ to $B$, while $\lambda=-1$ is $B$ reflected about $A$.
The main advantage of this representation is its structural invariance under coordinate transformations (whereas the slope/intercept form depends on axes and fails on vertical lines). This is important in the context of Misner, Thorne and Wheeler, where general relativity bends the local coordinates of spacetime. It lets them treat worldlines between events, among other things, in a unified manner.