Mathematics

Three points are collinear when they all lie on the same straight line, which means the vectors connecting them must be parallel. The key insight is that if $P$, $Q$, and $R$ are collinear, then $\overrightarrow{PQ}$ and $\overrightarrow{PR}$ must be scalar multiples of each other—one vector is just a stretched or shrunk version of the other. Find both vectors, set up the proportionality condition, and solve for $\lambda$.

Three points are collinear when they all lie on the same straight line, which means the vectors connecting them must be parallel. The key insight is that if $P$, $Q$, and $R$ are collinear, then $\overrightarrow{PQ}$ and $\overrightarrow{PR}$ must be scalar multiples of each other—one vector is just a stretched or shrunk version of the other. Find both vectors, set up the proportionality condition, and solve for $\lambda$.