The unit sphere in R^3 is arcwise connected

So I attempted to find the intersection of the unit sphere with a plane that contains any two arbitrary points $\displaystyle a, b$ on the sphere and the origin, but I only came up with a really convoluted expression that I am not even sure can be used to find a function $\displaystyle f: [0,1] \to \mathbb{R}^{3}$ such that $\displaystyle f(0) = a, f(1) = b$ and for all $\displaystyle t\in [0,1]$, $\displaystyle f(t)$ is on the unit sphere. Any help would be appreciated.