For any three vectors `veca, vecb and vecc` write value of the following `veca xx (vecb + vecc) + vecb xx (vecc + veca)+ vecc xx (veca + vecb)`

Answer

To solve the expression \( \vec{a} \times (\vec{b} + \vec{c}) + \vec{b} \times (\vec{c} + \vec{a}) + \vec{c} \times (\vec{a} + \vec{b}) \), we will expand and simplify it step by step.

### Step 1: Expand Each Cross Product
We start by expanding each term in the expression:

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

Now, we can combine these results:

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

### Step 2: Rearranging Terms
Next, we can rearrange the terms to group them:

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

### Step 3: Apply the Property of Cross Products
Using the property of cross products, where \( \vec{x} \times \vec{y} = -(\vec{y} \times \vec{x}) \):

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

Substituting these into our expression gives:

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

### Step 4: Simplifying the Expression
Now, we can see that each pair cancels out:

\[ 0 + 0 + 0 = 0 \]

Thus, the value of the entire expression is:

\[ \boxed{0} \]