Question #41887

Let the state space of a probabilistic system be mapped to $\mathbb{R}^3$. Three events $A, B,$ and $C$ are represented by the vectors $\vec{v}_A = x\hat{i} + 2\hat{j} - \hat{k}$, $\vec{v}_B = -\hat{i} + y\hat{j} + 2\hat{k}$, and $\vec{v}_C = 2\hat{i} - \hat{j} + z\hat{k}$ respectively, where $x, y, z \in \mathbb{R}$. The probability of any event $E$ represented by a vector $\vec{v}$ is defined as $P(E) = \vec{v} \cdot \vec{u}$, where $\vec{u} = \frac{1}{3}(\hat{i} + \hat{j} + \hat{k})$. It is given that $A, B,$ and $C$ are mutually exclusive and exhaustive events, the vectors $\vec{v}_A, \vec{v}_B, \vec{v}_C$ are linearly dependent, and $x^2 + y^2 + z^2 = 14$. The number of possible ordered triplets $(x, y, z)$ is: