1. **State the problem:** Compute the scalar triple product $[\hat{i},\hat{j},\hat{k}] \cdot [\hat{i} - \hat{j} + \hat{k}]$. 2. **Given vectors:** Let $\mathbf{a} = [\hat{i}, \hat{j}, \hat{k}] = [1,1,1]$ and $\mathbf{b} = [\hat{i} - \hat{j} + \hat{k}] = [1, -1, 1]$. 3. **Clarify vector components:** For scalar triple product, it is usually defined as $\mathbf{a} \cdot (\mathbf{b} \times \mathbf{c})$. 4. **From your setup, it looks like you want:** $[\hat{i}, \hat{j}, \hat{k}] \cdot [\hat{i} - \hat{j} + \hat{k}]$, interpreted as $$ (\hat{i} \times \hat{j}) \cdot (\hat{i} - \hat{j} + \hat{k}).$$ 5. **Calculate cross product $\hat{i} \times \hat{j}$:** $$ \hat{i} \times \hat{j} = \hat{k} = [0,0,1].$$ 6. **Dot this with $[1, -1, 1]$:** $$ [0,0,1] \cdot [1,-1,1] = 0\times1 + 0\times(-1) + 1\times1 = 1.$$ 7. **Final answer:** $$ \boxed{1}. $$