$\vec{a}$ and $\vec{b}$ are non-collinear vectors. For which $\lambda$ are the points with position vectors $\vec{a} + 2\vec{b}$, $3\vec{a} - \vec{b}$ and $\lambda\vec{a} + 8\vec{b}$ collinear?

Explanation

1. 01

   Call the points $A$, $B$, $C$: $\overrightarrow{AB} = 2\vec{a} - 3\vec{b}$ and $\overrightarrow{AC} = (\lambda - 1)\vec{a} + 6\vec{b}$.

2. 02

   Collinear means $\overrightarrow{AC} = \mu\overrightarrow{AB}$; the $\vec{b}$ parts give $\mu = -2$, so $\lambda - 1 = -4$ and $\lambda = -3$.