`(vec(d) + vec(a)).(vec(a) xx (vec(b) xx (vec(c) xx vec(d))))` simplifies to

Options

A. `(vec(b).vec(d))[vec(a) vec(c) vec(d)]`
B. `(vec(b).vec(c))[vec(a) vec(b) vec(d)]`
C. `(vec(b).vec(a))[vec(a) vec(b) vec(d)]`
D. none of these

Answer

To simplify the expression \((\vec{d} + \vec{a}) \cdot (\vec{a} \times (\vec{b} \times (\vec{c} \times \vec{d})))\), we can follow these steps:

### Step 1: Apply the Vector Triple Product Identity
We start with the expression \(\vec{b} \times (\vec{c} \times \vec{d})\). 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 \(\vec{x} = \vec{b}\), \(\vec{y} = \vec{c}\), and \(\vec{z} = \vec{d}\). Thus, we have:
\[ \vec{b} \times (\vec{c} \times \vec{d}) = (\vec{b} \cdot \vec{d}) \vec{c} - (\vec{b} \cdot \vec{c}) \vec{d} \]

### Step 2: Substitute Back into the Expression
Now, substitute this result back into our original expression:
\[ (\vec{d} + \vec{a}) \cdot \left(\vec{a} \times \left((\vec{b} \cdot \vec{d}) \vec{c} - (\vec{b} \cdot \vec{c}) \vec{d}\right)\right) \]

### Step 3: Distribute the Dot Product
Using the distributive property of the dot product:
\[ = (\vec{d} + \vec{a}) \cdot \left((\vec{b} \cdot \vec{d}) (\vec{a} \times \vec{c}) - (\vec{b} \cdot \vec{c}) (\vec{a} \times \vec{d})\right) \]
This expands to:
\[ = (\vec{d} + \vec{a}) \cdot \left((\vec{b} \cdot \vec{d}) \vec{a} \times \vec{c}\right) - (\vec{b} \cdot \vec{c}) (\vec{d} + \vec{a}) \cdot (\vec{a} \times \vec{d}) \]

### Step 4: Evaluate Each Dot Product
1. The first term:
\[ (\vec{d} + \vec{a}) \cdot \left((\vec{b} \cdot \vec{d}) \vec{a} \times \vec{c}\right) = (\vec{b} \cdot \vec{d}) \left(\vec{d} + \vec{a}\right) \cdot (\vec{a} \times \vec{c}) \]
2. The second term:
\[ (\vec{d} + \vec{a}) \cdot (\vec{a} \times \vec{d}) = 0 \]
This is because \(\vec{a} \times \vec{d}\) is perpendicular to both \(\vec{a}\) and \(\vec{d}\).

### Step 5: Final Expression
Thus, we are left with:
\[ (\vec{b} \cdot \vec{d}) (\vec{d} + \vec{a}) \cdot (\vec{a} \times \vec{c}) \]

### Step 6: Rearranging the Result
This can be written as:
\[ (\vec{b} \cdot \vec{d}) \cdot \text{(scalar triple product)} \quad \text{where the scalar triple product is } (\vec{a}, \vec{c}, \vec{d}). \]

### Final Answer
Thus, the simplified expression is:
\[ \vec{b} \cdot \vec{d} \cdot (\vec{a} \times \vec{c} \times \vec{d}) \]