Question #75996

Let $\vec{a} = \hat{i} + x\hat{j}$, $\vec{b} = \hat{j} + x\hat{k}$, and $\vec{c} = \hat{k} + x\hat{i}$ for $x \in \mathbb{R}$. Define the vectors $\vec{u} = \vec{a} \times (\vec{b} \times \vec{c})$, $\vec{v} = \vec{b} \times (\vec{c} \times \vec{a})$, and $\vec{w} = \vec{a} \times \vec{b}$. Let $M(x)$ be a $3 \times 3$ matrix whose first, second, and third rows are the components of the vectors $\vec{v}$, $\vec{u}$, and $\vec{w}$ respectively. If $f(x) = \det(M(x))$, evaluate the sum: $S = \sum_{k=1}^{100} \left[ f\left(\frac{k}{100}\right) \right]$ where $[ \cdot ]$ denotes the greatest integer function.