If the vectors `veca,vecb,vecc` form the sides BC,CA and AB respectively of a triangle ABC then (A) `veca.(vecbxxvecc)=vec0` (B) `vecaxx(vecbxvecc)=vec0` (C) `veca.vecb=vecc=vecc=veca.a!=0` (D) `vecaxxvecb+vecbxxvecc+veccxxvecavec0`

Options

A. `veca.vecb+ vecb.vecc+vecc.veca=0`
B. `vecaxxvecb = vecbxxvecc =veccxxveca`
C. `veca.vecb = vecb.vecc = vecc.veca`
D. `vecaxxvecb + vecbxxvecc +veccxxveca=vec0`

Explanation

To solve the problem, we need to analyze the properties of the vectors that represent the sides of triangle ABC. Given that the vectors \(\vec{a}, \vec{b}, \vec{c}\) represent the sides BC, CA, and AB respectively, we can derive the relationships between these vectors.

### Step-by-Step Solution:

1. **Understanding the Triangle Condition**:
Since \(\vec{a}, \vec{b}, \vec{c}\) are the sides of triangle ABC, we can use the triangle law of vectors which states that the sum of the vectors forming the triangle is equal to zero:
\[ \vec{a} + \vec{b} + \vec{c} = \vec{0} \]

2. **Rearranging the Equation**:
From the equation above, we can rearrange it to express one vector in terms of the others:
\[ \vec{c} = -(\vec{a} + \vec{b}) \]

3. **Taking the Cross Product**:
We can take the cross product of \(\vec{c}\) with \(\vec{a}\):
\[ \vec{c} \times \vec{a} = -(\vec{a} + \vec{b}) \times \vec{a} \]
Since \(\vec{a} \times \vec{a} = \vec{0}\), we simplify this to:
\[ \vec{c} \times \vec{a} = -\vec{b} \times \vec{a} \]

4. **Rearranging the Cross Product**:
We can rearrange the equation:
\[ \vec{c} \times \vec{a} + \vec{b} \times \vec{a} = \vec{0} \]
This implies:
\[ \vec{c} \times \vec{a} = -\vec{b} \times \vec{a} \]

5. **Taking Another Cross Product**:
Now, we take the cross product of \(\vec{b}\) with the original equation:
\[ \vec{b} \times (\vec{a} + \vec{b} + \vec{c}) = \vec{0} \]
This leads to:
\[ \vec{b} \times \vec{a} + \vec{b} \times \vec{b} + \vec{b} \times \vec{c} = \vec{0} \]
Again, since \(\vec{b} \times \vec{b} = \vec{0}\), we have:
\[ \vec{b} \times \vec{a} + \vec{b} \times \vec{c} = \vec{0} \]

6. **Final Relationships**:
From the above equations, we can conclude:
\[ \vec{a} \times \vec{b} + \vec{b} \times \vec{c} + \vec{c} \times \vec{a} = \vec{0} \]
This shows that the sum of the cross products of the sides of the triangle is equal to zero.

### Conclusion:
Thus, the correct option from the provided choices is:
**(D)** \(\vec{a} \times \vec{b} + \vec{b} \times \vec{c} + \vec{c} \times \vec{a} = \vec{0}\).