If `hata and hatb` are unit vectors then the vector defined as `vecV=(hata xx hatb) xx (hata+hatb)` is collinear to the vector :

Options

A. `hata+hatb`
B. `hatb-hata`
C. `2hata-hatb`
D. `hata+2hatb`

Answer

To solve the problem, we need to find the vector \(\vec{V} = (\hat{a} \times \hat{b}) \times (\hat{a} + \hat{b})\) and determine which vector it is collinear with.

### Step 1: Define the vectors
Let \(\hat{a}\) and \(\hat{b}\) be unit vectors. This means that:
\[ |\hat{a}| = 1 \quad \text{and} \quad |\hat{b}| = 1 \]

### Step 2: Rewrite the vector \(\vec{V}\)
We can express \(\vec{V}\) as:
\[ \vec{V} = (\hat{a} \times \hat{b}) \times (\hat{a} + \hat{b}) \]
Let \(\hat{m} = \hat{a} + \hat{b}\).

### Step 3: Use the vector triple product identity
The vector triple product identity states that:
\[ \vec{u} \times (\vec{v} \times \vec{w}) = (\vec{u} \cdot \vec{w}) \vec{v} - (\vec{u} \cdot \vec{v}) \vec{w} \]
Applying this to our expression:
\[ \vec{V} = (\hat{a} \times \hat{b}) \times \hat{m} = (\hat{a} \cdot \hat{m}) \hat{b} - (\hat{b} \cdot \hat{m}) \hat{a} \]

### Step 4: Calculate the dot products
Now we need to calculate \(\hat{a} \cdot \hat{m}\) and \(\hat{b} \cdot \hat{m}\):
\[ \hat{m} = \hat{a} + \hat{b} \]
Thus,
\[ \hat{a} \cdot \hat{m} = \hat{a} \cdot (\hat{a} + \hat{b}) = \hat{a} \cdot \hat{a} + \hat{a} \cdot \hat{b} = 1 + \hat{a} \cdot \hat{b} \]
And,
\[ \hat{b} \cdot \hat{m} = \hat{b} \cdot (\hat{a} + \hat{b}) = \hat{b} \cdot \hat{a} + \hat{b} \cdot \hat{b} = \hat{b} \cdot \hat{a} + 1 \]

### Step 5: Substitute back into \(\vec{V}\)
Now substituting back into the expression for \(\vec{V}\):
\[ \vec{V} = (1 + \hat{a} \cdot \hat{b}) \hat{b} - (\hat{b} \cdot \hat{a} + 1) \hat{a} \]
This simplifies to:
\[ \vec{V} = (1 + \hat{a} \cdot \hat{b}) \hat{b} - (1 + \hat{a} \cdot \hat{b}) \hat{a} \]
Factoring out \((1 + \hat{a} \cdot \hat{b})\):
\[ \vec{V} = (1 + \hat{a} \cdot \hat{b})(\hat{b} - \hat{a}) \]

### Step 6: Determine collinearity
The vector \(\vec{V}\) is collinear with \(\hat{b} - \hat{a}\) since it is a scalar multiple of \(\hat{b} - \hat{a}\).

### Conclusion
Thus, the vector \(\vec{V}\) is collinear to the vector \(\hat{b} - \hat{a}\).