1. **Problem:** Find the position vector of vertex D of parallelogram ABCD given position vectors of A, B, and C. 2. **Given:** $$A = 4\hat{i} + 2\hat{j} - 6\hat{k},\quad B = 5\hat{i} - 3\hat{j} + \hat{k},\quad C = 12\hat{i} + 4\hat{j} + 5\hat{k}$$ 3. **Formula:** In a parallelogram, the position vector of D is given by $$\vec{D} = \vec{A} + \vec{C} - \vec{B}$$ 4. **Calculation:** Calculate each component: $$\hat{i}: 4 + 12 - 5 = 11$$ $$\hat{j}: 2 + 4 - (-3) = 2 + 4 + 3 = 9$$ $$\hat{k}: -6 + 5 - 1 = -2$$ 5. **Result:** $$\vec{D} = 11\hat{i} + 9\hat{j} - 2\hat{k}$$ 6. **Answer:** Option c) $11 \hat{i} + 9 \hat{j} - 2 \hat{k}$ This means the position vector of vertex D is $11 \hat{i} + 9 \hat{j} - 2 \hat{k}$.