If `veca` and `vecb` are unit vectors, then which of the following values of `veca.vecb` is not possible ?

Options

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

Answer

To solve the problem, we need to determine which of the given values for the dot product of two unit vectors \(\vec{a}\) and \(\vec{b}\) is not possible.

### Step-by-Step Solution:

1. **Understanding Unit Vectors:**
- A unit vector is a vector that has a magnitude of 1. Therefore, for unit 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 \(\vec{a}\) and \(\vec{b}\) 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:**
- The cosine of an angle \(\theta\) can take values in the range:
\[ -1 \leq \cos \theta \leq 1 \]
- Therefore, the possible values for \(\vec{a} \cdot \vec{b}\) must also lie within this range:
\[ -1 \leq \vec{a} \cdot \vec{b} \leq 1 \]

4. **Evaluating Given Options:**
- Now, we will evaluate the given options:
- Option 1: \(\sqrt{3}\)
- Option 2: \(\frac{\sqrt{3}}{2}\)
- Option 3: \(\frac{1}{\sqrt{2}}\)
- Option 4: \(-\frac{1}{2}\)

- **Option 1: \(\sqrt{3}\)**
- Since \(\sqrt{3} \approx 1.732\), which is greater than 1, this value is not possible.

- **Option 2: \(\frac{\sqrt{3}}{2}\)**
- This value is approximately \(0.866\), which is within the range \([-1, 1]\).

- **Option 3: \(\frac{1}{\sqrt{2}}\)**
- This value is approximately \(0.707\), which is also within the range \([-1, 1]\).

- **Option 4: \(-\frac{1}{2}\)**
- This value is \(-0.5\), which is within the range \([-1, 1]\).

5. **Conclusion:**
- The only value that is not possible for \(\vec{a} \cdot \vec{b}\) is \(\sqrt{3}\). Therefore, the correct answer is:
\[ \text{Option 1: } \sqrt{3} \]