Question #41252

Let $\vec{a}, \vec{b}, \vec{c}$ be three non-coplanar vectors in $\mathbb{R}^3$. Three new vectors are defined as: $\vec{u} = \vec{a} + 2\vec{b} - \vec{c}$ $\vec{v} = 3\vec{a} + \lambda\vec{b} + \vec{c}$ $\vec{w} = \vec{a} + 4\vec{b} + \lambda\vec{c}$ where $\lambda \in \mathbb{R}$. A $3 \times 3$ matrix $M$ is constructed such that its elements are given by: $M = \begin{pmatrix} \vec{u}\cdot\vec{a} & \vec{u}\cdot\vec{b} & \vec{u}\cdot\vec{c} \\ \vec{v}\cdot\vec{a} & \vec{v}\cdot\vec{b} & \vec{v}\cdot\vec{c} \\ \vec{w}\cdot\vec{a} & \vec{w}\cdot\vec{b} & \vec{w}\cdot\vec{c} \end{pmatrix}$ If $M$ is a singular matrix, what is the absolute difference between the possible values of $\lambda$?