For any three vectors a, b, c
`(a-b).{(b-c) xx (c-a)} = 2a.(bxxc)`.
(a) True
(b) False.

Answer

To determine whether the statement \((\mathbf{a} - \mathbf{b}) \cdot ((\mathbf{b} - \mathbf{c}) \times (\mathbf{c} - \mathbf{a})) = 2\mathbf{a} \cdot (\mathbf{b} \times \mathbf{c})\) is true or false, we will analyze both sides of the equation step by step.

### Step 1: Expand the left-hand side
We start with the left-hand side of the equation:
\[ (\mathbf{a} - \mathbf{b}) \cdot ((\mathbf{b} - \mathbf{c}) \times (\mathbf{c} - \mathbf{a})) \]
Let \(\mathbf{u} = \mathbf{b} - \mathbf{c}\) and \(\mathbf{v} = \mathbf{c} - \mathbf{a}\). Then, we can rewrite the left-hand side as:
\[ (\mathbf{a} - \mathbf{b}) \cdot (\mathbf{u} \times \mathbf{v}) \]

### Step 2: Calculate the cross product
Next, we need to compute the cross product \(\mathbf{u} \times \mathbf{v}\):
\[ \mathbf{u} \times \mathbf{v} = (\mathbf{b} - \mathbf{c}) \times (\mathbf{c} - \mathbf{a}) \]
Using the distributive property of the cross product:
\[ \mathbf{u} \times \mathbf{v} = \mathbf{b} \times \mathbf{c} - \mathbf{b} \times \mathbf{a} - \mathbf{c} \times \mathbf{c} + \mathbf{c} \times \mathbf{a} \]
Since \(\mathbf{c} \times \mathbf{c} = \mathbf{0}\), we have:
\[ \mathbf{u} \times \mathbf{v} = \mathbf{b} \times \mathbf{c} - \mathbf{b} \times \mathbf{a} + \mathbf{c} \times \mathbf{a} \]

### Step 3: Substitute back into the dot product
Now substituting back into the dot product:
\[ (\mathbf{a} - \mathbf{b}) \cdot (\mathbf{b} \times \mathbf{c} - \mathbf{b} \times \mathbf{a} + \mathbf{c} \times \mathbf{a}) \]
This expands to:
\[ (\mathbf{a} - \mathbf{b}) \cdot (\mathbf{b} \times \mathbf{c}) - (\mathbf{a} - \mathbf{b}) \cdot (\mathbf{b} \times \mathbf{a}) + (\mathbf{a} - \mathbf{b}) \cdot (\mathbf{c} \times \mathbf{a}) \]

### Step 4: Analyze each term
1. The first term \((\mathbf{a} - \mathbf{b}) \cdot (\mathbf{b} \times \mathbf{c})\) remains as is.
2. The second term \((\mathbf{a} - \mathbf{b}) \cdot (\mathbf{b} \times \mathbf{a})\) is zero because \(\mathbf{b} \times \mathbf{a}\) is perpendicular to \((\mathbf{a} - \mathbf{b})\).
3. The third term \((\mathbf{a} - \mathbf{b}) \cdot (\mathbf{c} \times \mathbf{a})\) also evaluates to zero because \(\mathbf{c} \times \mathbf{a}\) is perpendicular to \((\mathbf{a} - \mathbf{b})\).

Thus, we simplify the left-hand side to:
\[ (\mathbf{a} - \mathbf{b}) \cdot (\mathbf{b} \times \mathbf{c}) \]

### Step 5: Compare with the right-hand side
Now we need to compare this with the right-hand side:
\[ 2\mathbf{a} \cdot (\mathbf{b} \times \mathbf{c}) \]
The left-hand side simplifies to:
\[ \mathbf{a} \cdot (\mathbf{b} \times \mathbf{c}) - \mathbf{b} \cdot (\mathbf{b} \times \mathbf{c}) = \mathbf{a} \cdot (\mathbf{b} \times \mathbf{c}) \]
Since \(\mathbf{b} \cdot (\mathbf{b} \times \mathbf{c}) = 0\).

### Conclusion
Thus, we have:
\[ \mathbf{a} \cdot (\mathbf{b} \times \mathbf{c}) \neq 2\mathbf{a} \cdot (\mathbf{b} \times \mathbf{c}) \]
Hence, the statement is **False**.