If `vec a and vec b` are unit vectors and `vec c` is such that `vec c` is such that `vec c = vec c = vec a xx vec c + vec b` then maximum value of `[veca vec b 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 the conditions in the question.

### Step-by-Step Solution:

1. **Understanding the Given Information**:
We have two unit vectors \(\vec{a}\) and \(\vec{b}\), and a vector \(\vec{c}\) defined by the equation:
\[ \vec{c} = \vec{a} \times \vec{c} + \vec{b} \]
This implies that \(\vec{c}\) is dependent on itself and the vectors \(\vec{a}\) and \(\vec{b}\).

2. **Rearranging the Equation**:
Rearranging the equation for \(\vec{c}\):
\[ \vec{c} - \vec{a} \times \vec{c} = \vec{b} \]
This can be rewritten as:
\[ \vec{c} (I - \vec{a} \times) = \vec{b} \]
where \(I\) is the identity operator.

3. **Taking Dot Products**:
We take the dot product of \(\vec{c}\) with both sides:
\[ \vec{c} \cdot \vec{c} - \vec{c} \cdot (\vec{a} \times \vec{c}) = \vec{c} \cdot \vec{b} \]
Since \(\vec{c} \cdot (\vec{a} \times \vec{c}) = 0\) (the dot product of a vector with a vector perpendicular to it is zero), we have:
\[ \|\vec{c}\|^2 = \vec{c} \cdot \vec{b} \]

4. **Finding the Maximum Value of the Scalar Triple Product**:
The scalar triple product \([ \vec{a}, \vec{b}, \vec{c} ]\) can be expressed as:
\[ [ \vec{a}, \vec{b}, \vec{c} ] = \vec{a} \cdot (\vec{b} \times \vec{c}) \]
To maximize this expression, we need to consider the geometric interpretation. The maximum value occurs when \(\vec{a}\), \(\vec{b}\), and \(\vec{c}\) are mutually perpendicular.

5. **Using the Properties of Unit Vectors**:
Since \(\vec{a}\) and \(\vec{b}\) are unit vectors, the maximum value of the scalar triple product is given by:
\[ \max |[ \vec{a}, \vec{b}, \vec{c} ]| = |\vec{a}| |\vec{b}| |\vec{c}| \sin(\theta) \]
where \(\theta\) is the angle between \(\vec{b}\) and \(\vec{c}\). The maximum occurs when \(\theta = 90^\circ\).

6. **Conclusion**:
Since \(\vec{a}\) and \(\vec{b}\) are unit vectors and \(\vec{c}\) can also be a unit vector under the right conditions, the maximum value of the scalar triple product is:
\[ \max |[ \vec{a}, \vec{b}, \vec{c} ]| = 1 \]