If `hata, hatb` are unit vectors and `vec c` is such that `vec c=veca xx vec c +vecb`, then the maximum value of `[veca vecb vec c]` is :

Options

A. 1
B. `(1)/(2)`
C. 2
D. `(3)/(2)`

Answer

To solve the problem, we need to find the maximum value of the scalar triple product \([ \vec{a}, \vec{b}, \vec{c} ]\) given that \(\vec{c} = \vec{a} \times \vec{c} + \vec{b}\) and that \(\vec{a}\) and \(\vec{b}\) are unit vectors.

### Step-by-Step Solution:

1. **Understanding the Given Equation:**
We start with the equation:
\[ \vec{c} = \vec{a} \times \vec{c} + \vec{b} \]

2. **Taking the Dot Product with \(\vec{a}\):**
We take the dot product of both sides with \(\vec{a}\):
\[ \vec{a} \cdot \vec{c} = \vec{a} \cdot (\vec{a} \times \vec{c}) + \vec{a} \cdot \vec{b} \]

3. **Using the Property of the Cross Product:**
The term \(\vec{a} \cdot (\vec{a} \times \vec{c})\) is zero because the dot product of a vector with a vector perpendicular to it is zero:
\[ \vec{a} \cdot \vec{c} = 0 + \vec{a} \cdot \vec{b} \]
Thus, we have:
\[ \vec{a} \cdot \vec{c} = \vec{a} \cdot \vec{b} \]

4. **Finding the Scalar Triple Product:**
The scalar triple product \([ \vec{a}, \vec{b}, \vec{c} ]\) can be expressed as:
\[ [ \vec{a}, \vec{b}, \vec{c} ] = \vec{a} \cdot (\vec{b} \times \vec{c}) \]

5. **Substituting for \(\vec{c}\):**
We substitute \(\vec{c}\) into the scalar triple product:
\[ [ \vec{a}, \vec{b}, \vec{c} ] = \vec{a} \cdot \left( \vec{b} \times (\vec{a} \times \vec{c} + \vec{b}) \right) \]

6. **Using the Vector Triple Product Identity:**
We can use the vector triple product identity:
\[ \vec{u} \times (\vec{v} \times \vec{w}) = (\vec{u} \cdot \vec{w}) \vec{v} - (\vec{u} \cdot \vec{v}) \vec{w} \]
Applying this, we get:
\[ [ \vec{a}, \vec{b}, \vec{c} ] = \vec{a} \cdot \left( \vec{b} \times \vec{b} + \vec{b} \times (\vec{a} \times \vec{c}) \right) \]
The first term is zero since \(\vec{b} \times \vec{b} = 0\):
\[ = \vec{a} \cdot \left( \vec{b} \times (\vec{a} \times \vec{c}) \right) \]

7. **Simplifying Further:**
Using the identity again:
\[ = \vec{a} \cdot \left( (\vec{a} \cdot \vec{c}) \vec{b} - (\vec{a} \cdot \vec{b}) \vec{c} \right) \]
Substitute \(\vec{a} \cdot \vec{c} = \vec{a} \cdot \vec{b}\):
\[ = (\vec{a} \cdot \vec{b}) \vec{a} \cdot \vec{b} - (\vec{a} \cdot \vec{b}) \vec{c} \]

8. **Finding Maximum Value:**
Since \(\vec{a}\) and \(\vec{b}\) are unit vectors, we have \(|\vec{a}| = 1\) and \(|\vec{b}| = 1\). Therefore, the maximum value of \(\vec{a} \cdot \vec{b}\) is 1 (when \(\vec{a}\) and \(\vec{b}\) are in the same direction).

9. **Conclusion:**
Thus, the maximum value of the scalar triple product \([ \vec{a}, \vec{b}, \vec{c} ]\) is:
\[ \text{Maximum value} = 1 \]