Question

If \( \mathbf{p} \) and \( \mathbf{q} \) are unit vectors, then which of the following values of \( \mathbf{p} \cdot \mathbf{q} \) is not possible?

• A. \( -\frac{1}{2} \)
• B. \( \frac{1}{\sqrt{2}} \)
• C. \( \frac{\sqrt{3}}{2} \)
• D. \( \sqrt{3} \)

Hint:

For unit vectors, the dot product \( \mathbf{p} \cdot \mathbf{q} \) must always lie between \( -1 \) and \( 1 \). Any value outside this range is impossible.

Answer 1

The Correct Option is D

To solve the problem, we need to find which value of \( \mathbf{p} \cdot \mathbf{q} \) is not possible given that \( \mathbf{p} \) and \( \mathbf{q} \) are unit vectors.

1. Understanding the Dot Product of Unit Vectors: 
The dot product of two vectors \( \mathbf{p} \) and \( \mathbf{q} \) is defined as:

\( \mathbf{p} \cdot \mathbf{q} = |\mathbf{p}| |\mathbf{q}| \cos \theta \)
Since both vectors are unit vectors, \( |\mathbf{p}| = 1 \) and \( |\mathbf{q}| = 1 \), so:

\( \mathbf{p} \cdot \mathbf{q} = \cos \theta \)
This means the value of the dot product must lie between \( -1 \) and \( 1 \), inclusive.

2. Evaluating Each Option:
(A) \( -\frac{1}{2} \) → Lies within the range [–1, 1] → Possible
(B) \( \frac{1}{\sqrt{2}} \approx 0.707 \) → Lies within the range [–1, 1] → Possible
(C) \( \frac{\sqrt{3}}{2} \approx 0.866 \) → Lies within the range [–1, 1] → Possible
(D) \( \sqrt{3} \approx 1.732 \) → Outside the allowed range → Not Possible

3. Conclusion:
The value \( \sqrt{3} \) is greater than 1 and hence not possible as a dot product of two unit vectors.

Final Answer:
The value of \( \mathbf{p} \cdot \mathbf{q} \) that is not possible is \( \sqrt{3} \).