If $\vec{a}$ and $\vec{b}$ are two unit vectors and $(\vec{a}+\vec{b}) imes(\vec{a} imes\vec{b})=\vec{c}$, then $\vec{c}$ is parallel to:

Question

If $\vec{a}$ and $\vec{b}$ are two unit vectors and $(\vec{a}+\vec{b})	imes(\vec{a}	imes\vec{b})=\vec{c}$, then $\vec{c}$ is parallel to:

Accepted Answer

To determine why Option 1 is the correct answer, we need to analyze the given vector expression: $(\vec{a}+\vec{b})\times(\vec{a}\times\vec{b})=\vec{c}$.

### Explanation:

1. **Vector Triple Product Identity**:
   The vector triple product identity states:
   $$
   \vec{u} \times (\vec{v} \times \vec{w}) = (\vec{u} \cdot \vec{w})\vec{v} - (\vec{u} \cdot \vec{v})\vec{w}
   $$
   Applying this identity to $(\vec{a}+\vec{b})\times(\vec{a}\times\vec{b})$, we let $\vec{u} = \vec{a} + \vec{b}$, $\vec{v} = \vec{a}$, and $\vec{w} = \vec{b}$.

2. **Applying the Identity**:
   $$
   (\vec{a}+\vec{b})\times(\vec{a}\times\vec{b}) = ((\vec{a}+\vec{b}) \cdot \vec{b})\vec{a} - ((\vec{a}+\vec{b}) \cdot \vec{a})\vec{b}
   $$

3. **Dot Products**:
   Since $\vec{a}$ and $\vec{b}$ are unit vectors, $\vec{a} \cdot \vec{a} = 1$ and $\vec{b} \cdot \vec{b} = 1$. Also, $\vec{a} \cdot \vec{b} = \cos \theta$, where $\theta$ is the angle between $\vec{a}$ and $\vec{b}$.
   $$
   (\vec{a}+\vec{b}) \cdot \vec{b} = \vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{b} = \cos \theta + 1
   $$
   $$
   (\vec{a}+\vec{b}) \cdot \vec{a} = \vec{a} \cdot \vec{a} + \vec{b} \cdot \vec{a} = 1 + \cos \theta
   $$

4. **Substituting Back**:
   $$
   \vec{c} = ((\cos \theta + 1)\vec{a} - (1 + \cos \theta)\vec{b})
   $$
   $$
   \vec{c} = \cos \theta \vec{a} + \vec{a} - \cos \theta \vec{b} - \vec{b}
   $$
   $$
   \vec{c} = (\vec{a} - \vec{b}) + \cos \theta (\vec{a} - \vec{b})
   $$
   $$
   \vec{c} = (1 + \cos \theta)(\vec{a} - \vec{b})
   $$

5. **Conclusion**:
   The expression $(1 + \cos \theta)(\vec{a} - \vec{b})$ shows that $\vec{c}$ is indeed parallel to $\vec{a} - \vec{b}$. Therefore, Option 1 is correct.

### Why Other Options Are Incorrect:

- **Option 2 ($\vec{a}+\vec{b}$)**: The expression derived shows $\vec{c}$ is parallel to $\vec{a} - \vec{b}$, not $\vec{a} + \vec{b}$.

- **Option 3 ($2\vec{a}-\vec{b}$)**: The derived expression does not match this form. The factor of 2 is not present in the correct expression.

- **Option 4 ($2\vec{a}+\vec{b}$)**: Similarly, the derived expression does not match this form. The vector $\vec{b}$ is subtracted, not added.

Thus, the correct answer is Option 1, as $\vec{c}$ is parallel to $\vec{a} - \vec{b}$.