Question #41201

Let $\vec{a}, \vec{b}, \vec{c}$ be three non-coplanar vectors representing the position vectors of points $A, B, C$ with respect to an origin $O$. A new origin $D$ is chosen with position vector $\vec{d} = x\vec{a} + y\vec{b} + z\vec{c}$ such that $x+y+z \neq 1$. Let $\vec{r}_A, \vec{r}_B, \vec{r}_C$ be the position vectors of $A, B, C$ with respect to $D$. A $3 \times 3$ matrix $M$ is formed such that its rows are the scalar components of the vectors $\vec{u}, \vec{v}, \vec{w}$ respectively, where: $\vec{u} = 7(\vec{r}_A \times \vec{r}_B) + 3(\vec{r}_B \times \vec{r}_C)$ $\vec{v} = 7(\vec{r}_B \times \vec{r}_C) + 7(\vec{r}_C \times \vec{r}_A)$ $\vec{w} = 3(\vec{r}_C \times \vec{r}_A) + \lambda(\vec{r}_B \times \vec{r}_A)$ If $M$ is a singular matrix for all such choices of $D$, then the value of $\lambda$ is: