If `vec( a)` and `vec(b)` are unit vectors, then while of the following values of `vec( a)`. `vec(b)` is not possible ?

Options

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

Answer

To solve the problem, we need to analyze the possible values of the dot product of two unit vectors \( \vec{a} \) and \( \vec{b} \).

### Step-by-Step Solution:

1. **Understanding Unit Vectors**:
- A unit vector has a magnitude of 1. Therefore, for vectors \( \vec{a} \) and \( \vec{b} \), we have:
\[ |\vec{a}| = 1 \quad \text{and} \quad |\vec{b}| = 1 \]

2. **Dot Product of Two Vectors**:
- The dot product of two vectors can be expressed as:
\[ \vec{a} \cdot \vec{b} = |\vec{a}| |\vec{b}| \cos \theta \]
- Since both \( \vec{a} \) and \( \vec{b} \) are unit vectors, this simplifies to:
\[ \vec{a} \cdot \vec{b} = 1 \cdot 1 \cdot \cos \theta = \cos \theta \]

3. **Range of Cosine Function**:
- The value of \( \cos \theta \) lies in the range:
\[ -1 \leq \cos \theta \leq 1 \]
- Therefore, the dot product \( \vec{a} \cdot \vec{b} \) must also satisfy:
\[ -1 \leq \vec{a} \cdot \vec{b} \leq 1 \]

4. **Evaluating Given Options**:
- We need to check which of the provided options falls outside the range of \([-1, 1]\):
- \( \frac{1}{\sqrt{2}} \approx 0.707 \) (valid)
- \( -\frac{1}{2} \) (valid)
- \( \frac{3}{2} \) (not valid, as it is greater than 1)
- \( \sqrt{3} \approx 1.732 \) (not valid, as it is greater than 1)

5. **Conclusion**:
- The values of \( \vec{a} \cdot \vec{b} \) that are not possible are \( \frac{3}{2} \) and \( \sqrt{3} \). Among the options given, the option that is not possible is \( \sqrt{3} \).

Thus, the answer is that the value of \( \vec{a} \cdot \vec{b} \) that is not possible is \( \sqrt{3} \).