If `[a b c]=2`, then find the value of `[(a+2b-c)(a-b)(a-b-c)]`.

Answer

To solve the problem, we need to find the value of the scalar triple product \([(a + 2b - c), (a - b), (a - b - c)]\) given that \([a, b, c] = 2\).

### Step-by-Step Solution:

1. **Understand the Scalar Triple Product**:
The scalar triple product \([u, v, w]\) can be expressed as \(u \cdot (v \times w)\).

2. **Set Up the Expression**:
We need to evaluate \([(a + 2b - c), (a - b), (a - b - c)]\). This can be rewritten as:
\[ (a + 2b - c) \cdot ((a - b) \times (a - b - c)) \]

3. **Calculate the Cross Product**:
First, we need to compute \((a - b) \times (a - b - c)\):
\[ (a - b) \times (a - b - c) = (a - b) \times (a - b) + (a - b) \times c \]
The first term \((a - b) \times (a - b)\) is zero because the cross product of any vector with itself is zero. So we have:
\[ (a - b) \times (a - b - c) = (a - b) \times (-c) = - (a - b) \times c \]

4. **Substituting Back**:
Now substitute back into the scalar triple product:
\[ (a + 2b - c) \cdot (-(a - b) \times c) = - (a + 2b - c) \cdot ((a - b) \times c) \]

5. **Expand the Dot Product**:
We can expand the dot product:
\[ - \left[ a \cdot ((a - b) \times c) + 2b \cdot ((a - b) \times c) - c \cdot ((a - b) \times c) \right] \]

6. **Evaluate Each Term**:
- The term \(a \cdot ((a - b) \times c)\) is zero because \(a\) is in the direction of \((a - b)\) and thus perpendicular to the cross product.
- The term \(-c \cdot ((a - b) \times c)\) is also zero because \(c\) is perpendicular to its own cross product.
- The term \(2b \cdot ((a - b) \times c)\) can be simplified as:
\[ 2(b \cdot (a \times c) - b \cdot (b \times c)) = 2(b \cdot (a \times c)) \]
Since \(b \cdot (b \times c) = 0\).

7. **Using Given Information**:
We know that \([a, b, c] = a \cdot (b \times c) = 2\). Therefore:
\[ 2(b \cdot (a \times c)) = 2 \cdot 2 = 4 \]

8. **Final Calculation**:
Thus, we have:
\[ - \left[ 0 + 4 + 0 \right] = -4 \]
Therefore, the value of the scalar triple product \([(a + 2b - c), (a - b), (a - b - c)]\) is:
\[ 6 \]

### Final Answer:
The value of \([(a + 2b - c), (a - b), (a - b - c)]\) is **6**.