`(a + 2b-c).[(a-b) xx (a-b-c)]` is equal to

Options

A. `[abc]`
B. `2[abc]`
C. `3[abc]`
D. none of these

Answer

To solve the expression \((a + 2b - c) \cdot [(a - b) \times (a - b - c)]\), we will follow these steps:

### Step 1: Simplify the Cross Product
First, we need to simplify the cross product \((a - b) \times (a - b - c)\).

Let:
\[ u = a - b \]
\[ v = a - b - c \]
Then,
\[ v = u - c \]
Thus, the cross product becomes:
\[ u \times v = (a - b) \times ((a - b) - c) = (a - b) \times (u - c) \]

### Step 2: Apply the Distributive Property
Using the distributive property of the cross product:
\[ u \times v = u \times u - u \times c \]
Since the cross product of any vector with itself is zero:
\[ u \times u = 0 \]
Thus, we have:
\[ u \times v = 0 - (a - b) \times c = -(a - b) \times c \]

### Step 3: Substitute Back into the Dot Product
Now, substitute this back into the original expression:
\[ (a + 2b - c) \cdot [-(a - b) \times c] \]
This simplifies to:
\[ -(a + 2b - c) \cdot [(a - b) \times c] \]

### Step 4: Expand the Dot Product
Now, we expand the dot product:
\[ -(a \cdot [(a - b) \times c] + 2b \cdot [(a - b) \times c] - c \cdot [(a - b) \times c]) \]

### Step 5: Analyze Each Term
1. The term \(a \cdot [(a - b) \times c]\) is a scalar triple product and equals zero because the first vector \(a\) is repeated.
2. The term \(2b \cdot [(a - b) \times c]\) can be simplified further.
3. The term \(-c \cdot [(a - b) \times c]\) is also zero for the same reason as the first term.

### Step 6: Resulting Expression
Thus, we are left with:
\[ -(0 + 2b \cdot [(a - b) \times c] - 0) = -2b \cdot [(a - b) \times c] \]

### Step 7: Final Simplification
Finally, we can express this as:
\[ -2(b \cdot [(a - b) \times c]) \]
This is a scalar triple product, which can be rearranged to:
\[ -2 \cdot (b \cdot (a \times c) - b \cdot (b \times c)) \]

### Conclusion
The final answer is:
\[ -2 \cdot (b \cdot (a \times c)) \]