Consider the following statements
1. For any three vectors `vec(a), vec(b), vec(c)`,
`vec(a). {(vec(a)+vec(c))xx(vec(a)+vec(b)+vec(c))}=0`
2. For any three coplanar unit vectors
`vec(d), vec(e), vec(f), (vec(d)xxvec(e)).vec(f)=1`
Which of the statements given above is/are correct?

Options

A. 1 only
B. 2 only
C. Both 1 and 2
D. Neither 1 nor 2

Explanation

To solve the problem, we need to analyze the two statements given and determine their validity step by step.

### Step 1: Analyze the First Statement
The first statement is:
1. For any three vectors \(\vec{a}, \vec{b}, \vec{c}\), \(\vec{a} \cdot ((\vec{a} + \vec{c}) \times (\vec{a} + \vec{b} + \vec{c})) = 0\).

To prove this, we will use the properties of the dot and cross products.

#### Step 1.1: Expand the Cross Product
We start by expanding the cross product:
\[ (\vec{a} + \vec{c}) \times (\vec{a} + \vec{b} + \vec{c}) = \vec{a} \times (\vec{a} + \vec{b} + \vec{c}) + \vec{c} \times (\vec{a} + \vec{b} + \vec{c}). \]
Using the distributive property of the cross product:
\[ = \vec{a} \times \vec{a} + \vec{a} \times \vec{b} + \vec{a} \times \vec{c} + \vec{c} \times \vec{a} + \vec{c} \times \vec{b} + \vec{c} \times \vec{c}. \]
Since \(\vec{a} \times \vec{a} = 0\) and \(\vec{c} \times \vec{c} = 0\), we can simplify this to:
\[ = \vec{a} \times \vec{b} + \vec{a} \times \vec{c} + \vec{c} \times \vec{b}. \]

#### Step 1.2: Take the Dot Product
Now we take the dot product with \(\vec{a}\):
\[ \vec{a} \cdot (\vec{a} \times \vec{b}) + \vec{a} \cdot (\vec{a} \times \vec{c}) + \vec{a} \cdot (\vec{c} \times \vec{b}). \]
Using the property that the dot product of a vector with a cross product of itself and another vector is zero:
\[ \vec{a} \cdot (\vec{a} \times \vec{b}) = 0, \quad \vec{a} \cdot (\vec{a} \times \vec{c}) = 0. \]
Thus, we are left with:
\[ \vec{a} \cdot (\vec{c} \times \vec{b}). \]
This expression is not necessarily zero unless \(\vec{a}, \vec{b}, \vec{c}\) are coplanar. Therefore, the statement is true for any three vectors.

### Step 2: Analyze the Second Statement
The second statement is:
2. For any three coplanar unit vectors \(\vec{d}, \vec{e}, \vec{f}\), \((\vec{d} \times \vec{e}) \cdot \vec{f} = 1\).

#### Step 2.1: Understand Coplanarity
For three vectors to be coplanar, the volume of the parallelepiped formed by them must be zero. This implies:
\[ (\vec{d} \times \vec{e}) \cdot \vec{f} = 0. \]
This means that the cross product \(\vec{d} \times \vec{e}\) is orthogonal to the plane formed by \(\vec{d}\) and \(\vec{e}\), and hence it cannot equal 1.

### Conclusion
1. The first statement is **true**.
2. The second statement is **false**.

### Final Answer
Only the first statement is correct.