The scalar `vecA.(vecB+vecC)xx(vecA+vecB+vecC)` equals (A) 0 (B) `[vecA vecB vecC]+[vecB vecC vecA]` (C) `[vecA vecB vecC]` (D) none of these

Options

A. 0
B. `[vecA vecB vecC]+ [vecB vecC vecA] `
C. `[vecA vecB vecC]`
D. none of these

Explanation

To solve the problem, we need to evaluate the expression \( \vec{A} \cdot (\vec{B} + \vec{C}) \times (\vec{A} + \vec{B} + \vec{C}) \).

### Step 1: Expand the expression
We start by expanding the expression using the distributive property of the dot product and the cross product.

\[ \vec{A} \cdot (\vec{B} + \vec{C}) \times (\vec{A} + \vec{B} + \vec{C}) = \vec{A} \cdot \left( \vec{B} \times (\vec{A} + \vec{B} + \vec{C}) + \vec{C} \times (\vec{A} + \vec{B} + \vec{C}) \right) \]

### Step 2: Apply the cross product
Now we can apply the cross product to each term separately:

\[ = \vec{A} \cdot \left( \vec{B} \times \vec{A} + \vec{B} \times \vec{B} + \vec{B} \times \vec{C} + \vec{C} \times \vec{A} + \vec{C} \times \vec{B} + \vec{C} \times \vec{C} \right) \]

### Step 3: Simplify using properties of cross product
We know that the cross product of any vector with itself is zero, i.e., \( \vec{B} \times \vec{B} = 0 \) and \( \vec{C} \times \vec{C} = 0 \). Thus, we can simplify:

\[ = \vec{A} \cdot \left( \vec{B} \times \vec{A} + \vec{B} \times \vec{C} + \vec{C} \times \vec{A} + \vec{C} \times \vec{B} \right) \]

### Step 4: Evaluate the dot products
Now we can evaluate the dot products:

1. \( \vec{A} \cdot (\vec{B} \times \vec{A}) = 0 \) (since the dot product of a vector with the cross product of itself and another vector is zero).
2. \( \vec{A} \cdot (\vec{C} \times \vec{C}) = 0 \) (same reasoning).
3. \( \vec{A} \cdot (\vec{B} \times \vec{C}) \) remains as is.
4. \( \vec{A} \cdot (\vec{C} \times \vec{B}) \) also remains as is.

Thus, we have:

\[ = 0 + 0 + \vec{A} \cdot (\vec{B} \times \vec{C}) + \vec{A} \cdot (\vec{C} \times \vec{B}) \]

### Step 5: Use the scalar triple product property
Using the property of scalar triple product, we know that \( \vec{A} \cdot (\vec{B} \times \vec{C}) = \vec{B} \cdot (\vec{C} \times \vec{A}) \). Therefore, we can simplify:

\[ = 0 + 0 = 0 \]

### Conclusion
Thus, the final answer is:

\[ \vec{A} \cdot (\vec{B} + \vec{C}) \times (\vec{A} + \vec{B} + \vec{C}) = 0 \]

### Final Answer
The correct option is (A) 0.

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