Question #40594

Let $S = \{\vec{v}_1, \vec{v}_2, \dots, \vec{v}_n\}$ be a set of vectors where $\vec{v}_k = (2k-1)\hat{i} + 4\hat{j} + (8-k)\hat{k}$. The total number of ways to select a non-empty subset of vectors from $S$ is $1023$. Let $\Pi_1$ be the plane containing the vectors $\vec{v}_1$ and $\vec{v}_2$, and $\Pi_2$ be the plane containing the vectors $\vec{v}_3$ and $\vec{v}_4$. If $\theta$ is the acute angle between the planes $\Pi_1$ and $\Pi_2$, evaluate the expression: $E = \frac{\cos \theta}{\binom{n}{2}} \sum_{1 \le i < j \le n} \frac{|\vec{v}_i|^2 |\vec{v}_j|^2 - (\vec{v}_i \cdot \vec{v}_j)^2}{(j-i)^2} + \sin \theta \sum_{k=1}^n |\vec{v}_k|^4$