Consider the following statements
1. For any three vectors `vec(a) vec(b) vec(c ) vec(a).{(vec(b) + vec(c )) xx vec(a) + vec(b) + vec(c )}=0`
2. for any three coplanar vectors `vec(d), vec(e ), vec(f), (vec(d) xx vec(e )) .vec(f )= 0`

Options

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

Answer

To solve the given problem, we need to analyze both statements regarding vectors and determine their validity.

### Step 1: Analyze the First Statement

The first statement is:
\[ \vec{a} \cdot \left( \vec{b} + \vec{c} \right) \times \left( \vec{a} + \vec{b} + \vec{c} \right) = 0 \]

1. **Expand the Cross Product:**
We can expand the expression:
\[ \vec{b} + \vec{c} \text{ cross } \left( \vec{a} + \vec{b} + \vec{c} \right) \]
This results in:
\[ (\vec{b} + \vec{c}) \times \vec{a} + (\vec{b} + \vec{c}) \times \vec{b} + (\vec{b} + \vec{c}) \times \vec{c} \]

2. **Calculate Each Term:**
- The term \((\vec{b} + \vec{c}) \times \vec{b}\) equals \(\vec{0}\) (since any vector crossed with itself is zero).
- The term \((\vec{b} + \vec{c}) \times \vec{c}\) also equals \(\vec{0}\).
- Thus, we are left with:
\[ \vec{a} \cdot \left( \vec{b} \times \vec{a} + \vec{c} \times \vec{a} \right) \]

3. **Evaluate the Dot Product:**
The dot product of a vector with the cross product of itself and another vector is always zero:
\[ \vec{a} \cdot (\vec{b} \times \vec{a}) = 0 \quad \text{and} \quad \vec{a} \cdot (\vec{c} \times \vec{a}) = 0 \]
Therefore, the entire expression equals zero.

### Conclusion for First Statement:
The first statement is **true**.

### Step 2: Analyze the Second Statement

The second statement is:
\[ (\vec{d} \times \vec{e}) \cdot \vec{f} = 0 \]

1. **Understanding Coplanarity:**
For three vectors \(\vec{d}, \vec{e}, \vec{f}\) to be coplanar, the volume of the parallelepiped they form must be zero. This is equivalent to saying that the scalar triple product is zero.

2. **Evaluate the Cross Product:**
The cross product \(\vec{d} \times \vec{e}\) gives a vector that is perpendicular to the plane formed by \(\vec{d}\) and \(\vec{e}\).

3. **Dot Product with \(\vec{f}\):**
Since \(\vec{f}\) lies in the same plane as \(\vec{d}\) and \(\vec{e}\), the dot product of \(\vec{f}\) with the vector \(\vec{d} \times \vec{e}\) (which is perpendicular to the plane) must be zero:
\[ (\vec{d} \times \vec{e}) \cdot \vec{f} = 0 \]

### Conclusion for Second Statement:
The second statement is **true**.

### Final Conclusion:
- The first statement is false.
- The second statement is true.

### Summary of Steps:
1. Expand the expression in the first statement and simplify.
2. Use properties of dot and cross products to evaluate the first statement.
3. Understand the implications of coplanarity in the second statement.
4. Conclude the validity of both statements.