`(i)a xx (b + c) + b xx (c + a) + c xx(a +b) =`

Answer

To solve the given vector expression step by step, we will use the properties of the cross product. The expression we need to simplify is:

\[ \mathbf{a} \times (\mathbf{b} + \mathbf{c}) + \mathbf{b} \times (\mathbf{c} + \mathbf{a}) + \mathbf{c} \times (\mathbf{a} + \mathbf{b}) \]

### Step 1: Expand the Cross Products
Using the distributive property of the cross product, we can expand each term in the expression:

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

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

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

### Step 2: Combine All Terms
Now, we can combine all the expanded terms together:

\[ (\mathbf{a} \times \mathbf{b} + \mathbf{a} \times \mathbf{c}) + (\mathbf{b} \times \mathbf{c} + \mathbf{b} \times \mathbf{a}) + (\mathbf{c} \times \mathbf{a} + \mathbf{c} \times \mathbf{b}) \]

This simplifies to:

\[ \mathbf{a} \times \mathbf{b} + \mathbf{a} \times \mathbf{c} + \mathbf{b} \times \mathbf{c} + \mathbf{b} \times \mathbf{a} + \mathbf{c} \times \mathbf{a} + \mathbf{c} \times \mathbf{b} \]

### Step 3: Rearranging Terms
We can rearrange the terms to group them by pairs:

\[ \mathbf{a} \times \mathbf{b} + \mathbf{b} \times \mathbf{a} + \mathbf{a} \times \mathbf{c} + \mathbf{c} \times \mathbf{a} + \mathbf{b} \times \mathbf{c} + \mathbf{c} \times \mathbf{b} \]

### Step 4: Apply the Antisymmetry Property
Using the property of the cross product that states \(\mathbf{x} \times \mathbf{y} = -(\mathbf{y} \times \mathbf{x})\), we can simplify:

1. \(\mathbf{b} \times \mathbf{a} = -(\mathbf{a} \times \mathbf{b})\)
2. \(\mathbf{c} \times \mathbf{a} = -(\mathbf{a} \times \mathbf{c})\)
3. \(\mathbf{c} \times \mathbf{b} = -(\mathbf{b} \times \mathbf{c})\)

Substituting these into the expression gives:

\[ \mathbf{a} \times \mathbf{b} - \mathbf{a} \times \mathbf{b} + \mathbf{a} \times \mathbf{c} - \mathbf{a} \times \mathbf{c} + \mathbf{b} \times \mathbf{c} - \mathbf{b} \times \mathbf{c} \]

### Step 5: Simplifying the Expression
All terms cancel out:

\[ 0 \]

### Final Result
Thus, the final result of the expression is:

\[ \boxed{0} \]