If `veca,vecb,vecc `are three vectors such that` |vec b∣=|vecc|` then `{(veca+vecb)×(veca+vecc)}×{(veca+vecb)×(veca+vecc)}×{(vecb×vecc).(vecb+vecc)}=`

Options

A. 0
B. 1
C. 2
D. 3

Answer

To solve the problem, we need to evaluate the expression:

\[ \{(\vec{a} + \vec{b}) \times (\vec{a} + \vec{c})\} \times \{(\vec{b} \times \vec{c}) \cdot (\vec{b} + \vec{c})\} \]

Given that \(|\vec{b}| = |\vec{c}|\), we can proceed with the solution step by step.

### Step 1: Simplify the Cross Product
First, we simplify the expression \((\vec{a} + \vec{b}) \times (\vec{a} + \vec{c})\).

Using the distributive property of the cross product:

\[ (\vec{a} + \vec{b}) \times (\vec{a} + \vec{c}) = \vec{a} \times \vec{a} + \vec{a} \times \vec{c} + \vec{b} \times \vec{a} + \vec{b} \times \vec{c} \]

Since \(\vec{a} \times \vec{a} = \vec{0}\), this simplifies to:

\[ \vec{a} \times \vec{c} + \vec{b} \times \vec{a} + \vec{b} \times \vec{c} \]

### Step 2: Cross Product with Itself
Next, we need to evaluate the expression:

\[ \{(\vec{a} + \vec{b}) \times (\vec{a} + \vec{c})\} \times \{(\vec{b} \times \vec{c}) \cdot (\vec{b} + \vec{c})\} \]

Since we are taking the cross product of a vector with itself, we know that:

\[ \vec{u} \times \vec{u} = \vec{0} \]

Thus, if we denote \( \vec{p} = (\vec{a} + \vec{b}) \times (\vec{a} + \vec{c}) \), we can see that:

\[ \vec{p} \times \vec{p} = \vec{0} \]

### Step 3: Final Expression
Now, we can conclude that the entire expression evaluates to:

\[ \vec{0} \times \{(\vec{b} \times \vec{c}) \cdot (\vec{b} + \vec{c})\} = \vec{0} \]

### Conclusion
Therefore, the final result of the expression is:

\[ \boxed{0} \]