Value of `[vec a xx vec b,vec a xx vecc,vec d]` is always equal to (a) ( ⃗ a . ⃗ d ) [ ⃗ a ⃗ b ⃗ c ] (b) ( ⃗ a . ⃗ c ) [ ⃗ a ⃗ b ⃗ d ] (c) ( ⃗ a . ⃗ b ) [ ⃗ a ⃗ b ⃗ d ] (d) none of these

Options

A. `(veca.vecd) [vecavecbvecc]`
B. `(veca.vecc)[veca vecb vecd]
C. `(veca.vecb)[veca vecb vecd]`
D. none of these

Explanation

To solve the problem, we need to evaluate the expression \([ \vec{a} \times \vec{b}, \vec{a} \times \vec{c}, \vec{d} ]\) and determine its value.

### Step-by-Step Solution:

1. **Understanding the Expression**:
The expression \([ \vec{a} \times \vec{b}, \vec{a} \times \vec{c}, \vec{d} ]\) represents the scalar triple product of three vectors: \(\vec{a} \times \vec{b}\), \(\vec{a} \times \vec{c}\), and \(\vec{d}\).

2. **Using the Scalar Triple Product Property**:
The scalar triple product can be expressed as:
\[ [\vec{u}, \vec{v}, \vec{w}] = \vec{u} \cdot (\vec{v} \times \vec{w}) \]
In our case, we can rewrite the expression as:
\[ [ \vec{a} \times \vec{b}, \vec{a} \times \vec{c}, \vec{d} ] = \vec{d} \cdot ((\vec{a} \times \vec{b}) \times (\vec{a} \times \vec{c})) \]

3. **Applying the Vector Triple Product Identity**:
We can use the vector triple product identity:
\[ \vec{x} \times (\vec{y} \times \vec{z}) = (\vec{x} \cdot \vec{z}) \vec{y} - (\vec{x} \cdot \vec{y}) \vec{z} \]
Let \(\vec{x} = \vec{a}\), \(\vec{y} = \vec{b}\), and \(\vec{z} = \vec{c}\). Thus, we have:
\[ (\vec{a} \times \vec{b}) \times (\vec{a} \times \vec{c}) = (\vec{a} \cdot \vec{c}) \vec{b} - (\vec{a} \cdot \vec{b}) \vec{c} \]

4. **Substituting Back**:
Now substituting this back into our expression gives:
\[ [ \vec{a} \times \vec{b}, \vec{a} \times \vec{c}, \vec{d} ] = \vec{d} \cdot \left( (\vec{a} \cdot \vec{c}) \vec{b} - (\vec{a} \cdot \vec{b}) \vec{c} \right) \]

5. **Distributing the Dot Product**:
Distributing the dot product, we have:
\[ [ \vec{a} \times \vec{b}, \vec{a} \times \vec{c}, \vec{d} ] = (\vec{a} \cdot \vec{c})(\vec{d} \cdot \vec{b}) - (\vec{a} \cdot \vec{b})(\vec{d} \cdot \vec{c}) \]

6. **Identifying the Options**:
Now we can analyze the options given in the question:
- (a) \((\vec{a} \cdot \vec{d})[\vec{a} \vec{b} \vec{c}]\)
- (b) \((\vec{a} \cdot \vec{c})[\vec{a} \vec{b} \vec{d}]\)
- (c) \((\vec{a} \cdot \vec{b})[\vec{a} \vec{b} \vec{d}]\)
- (d) none of these

The expression we derived does not match any of the options directly, thus the answer is **(d) none of these**.