The value of `[(a-b).(b-c)xx(c-a)]` is

Options

A. 0
B. `2[a,b,c]`
C. `[a,b,c]`
D. None

Answer

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

### Step 1: Understand the Expression
We have a dot product between the vector \((a-b)\) and the cross product \((b-c) \times (c-a)\).

### Step 2: Calculate the Cross Product
First, we need to compute the cross product \((b-c) \times (c-a)\).

Using the properties of cross products:
\[ (b-c) \times (c-a) = b \times c - b \times a - c \times c + c \times a \]
Since the cross product of any vector with itself is zero, \(c \times c = 0\). Thus, we simplify it to:
\[ (b-c) \times (c-a) = b \times c - b \times a + c \times a \]

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

### Step 4: Distribute the Dot Product
Now we distribute the dot product:
\[ = (a-b) \cdot (b \times c) - (a-b) \cdot (b \times a) + (a-b) \cdot (c \times a) \]

### Step 5: Simplify Each Term
1. **First Term**: \((a-b) \cdot (b \times c)\) is a scalar triple product which can be simplified based on the properties of the dot and cross products.
2. **Second Term**: \((a-b) \cdot (b \times a) = 0\) because \(b \times a\) is perpendicular to both \(b\) and \(a\).
3. **Third Term**: \((a-b) \cdot (c \times a)\) is also a scalar triple product.

### Step 6: Final Expression
Putting it all together, we find that the expression simplifies to:
\[ = (a-b) \cdot (b \times c) + (a-b) \cdot (c \times a) \]

### Step 7: Conclusion
The value of \([(a-b) \cdot ((b-c) \times (c-a))]\) is determined by the scalar triple products, which can be evaluated based on the specific vectors involved.