for any three vectors, `veca, vecb and vecc , (veca-vecb) . (vecb -vecc) xx (vecc -veca) = 2 veca.vecb xx vecc`.

Answer

To solve the problem, we need to verify the equation:

\[ (\vec{a} - \vec{b}) \cdot (\vec{b} - \vec{c}) \times (\vec{c} - \vec{a}) = 2 \vec{a} \cdot (\vec{b} \times \vec{c}) \]

### Step 1: Rewrite the left-hand side

We start with the left-hand side:

\[ (\vec{a} - \vec{b}) \cdot (\vec{b} - \vec{c}) \times (\vec{c} - \vec{a}) \]

### Step 2: Use the properties of the dot and cross products

We can rewrite the expression using the distributive property of the dot product:

\[ = \vec{a} \cdot ((\vec{b} - \vec{c}) \times (\vec{c} - \vec{a})) - \vec{b} \cdot ((\vec{b} - \vec{c}) \times (\vec{c} - \vec{a})) \]

### Step 3: Expand the cross product

Now, we need to compute the cross product \((\vec{b} - \vec{c}) \times (\vec{c} - \vec{a})\):

Using the distributive property of the cross product:

\[ (\vec{b} - \vec{c}) \times (\vec{c} - \vec{a}) = \vec{b} \times \vec{c} - \vec{b} \times \vec{a} - \vec{c} \times \vec{c} + \vec{c} \times \vec{a} \]

Since \(\vec{c} \times \vec{c} = \vec{0}\), we have:

\[ = \vec{b} \times \vec{c} - \vec{b} \times \vec{a} + \vec{c} \times \vec{a} \]

### Step 4: Substitute back into the expression

Substituting this back into our expression gives:

\[ = \vec{a} \cdot (\vec{b} \times \vec{c} - \vec{b} \times \vec{a} + \vec{c} \times \vec{a}) - \vec{b} \cdot (\vec{b} \times \vec{c} - \vec{b} \times \vec{a} + \vec{c} \times \vec{a}) \]

### Step 5: Simplify the dot products

Using the property that \(\vec{x} \cdot (\vec{y} \times \vec{z})\) gives the volume of the parallelepiped formed by the vectors, we can simplify:

1. \(\vec{a} \cdot (\vec{b} \times \vec{c})\)
2. \(\vec{a} \cdot (\vec{b} \times \vec{a}) = 0\) (since \(\vec{a} \times \vec{a} = \vec{0}\))
3. \(\vec{a} \cdot (\vec{c} \times \vec{a}) = 0\) (for the same reason)

For the second term:

1. \(\vec{b} \cdot (\vec{b} \times \vec{c}) = 0\)
2. \(\vec{b} \cdot (\vec{b} \times \vec{a}) = 0\)
3. \(\vec{b} \cdot (\vec{c} \times \vec{a})\)

Thus, we have:

\[ = \vec{a} \cdot (\vec{b} \times \vec{c}) - \vec{b} \cdot (\vec{c} \times \vec{a}) \]

### Step 6: Final simplification

Now, we can see that the left-hand side simplifies to:

\[ = \vec{a} \cdot (\vec{b} \times \vec{c}) - \vec{b} \cdot (\vec{c} \times \vec{a}) \]

### Step 7: Check the equality

Now we need to check if this equals \(2 \vec{a} \cdot (\vec{b} \times \vec{c})\).

From our simplification, we have:

\[ \vec{a} \cdot (\vec{b} \times \vec{c}) - \vec{b} \cdot (\vec{c} \times \vec{a}) \neq 2 \vec{a} \cdot (\vec{b} \times \vec{c}) \]

Thus, the original statement is **false**.