1. The problem asks to write the vector $(0,0,0,0,0,0,0,0,0,10)$ in builder notation. 2. Builder notation for vectors typically expresses the vector as a linear combination of unit vectors $\mathbf{e}_i$ where each $\mathbf{e}_i$ has a 1 in the $i$-th position and 0 elsewhere. 3. Here, the vector has all zeros except for the 10 in the 10th position. 4. Therefore, the vector can be written as $$10 \mathbf{e}_{10}$$ where $\mathbf{e}_{10} = (0,0,0,0,0,0,0,0,0,1)$. 5. This means the vector is 10 times the 10th unit vector. Final answer: $$\boxed{10 \mathbf{e}_{10}}$$