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 determine the impossible value of \( \mathbf{p} \cdot \mathbf{q} \), given that \( \mathbf{p} \) and \( \mathbf{q} \) are unit vectors.

1. Dot Product of Unit Vectors:
The dot product \( \mathbf{p} \cdot \mathbf{q} \) is defined as \( |\mathbf{p}| |\mathbf{q}| \cos \theta \). Since \( |\mathbf{p}| = 1 \) and \( |\mathbf{q}| = 1 \) for unit vectors, the dot product simplifies to \( \mathbf{p} \cdot \mathbf{q} = \cos \theta \). Therefore, the value of the dot product must be within the range \( [-1, 1] \).

2. Option Evaluation:
(A) \( -\frac{1}{2} \) is in \( [-1, 1] \) → Possible.
(B) \( \frac{1}{\sqrt{2}} \approx 0.707 \) is in \( [-1, 1] \) → Possible.
(C) \( \frac{\sqrt{3}}{2} \approx 0.866 \) is in \( [-1, 1] \) → Possible.
(D) \( \sqrt{3} \approx 1.732 \) is outside \( [-1, 1] \) → Not Possible.

3. Conclusion:
The value \( \sqrt{3} \) exceeds 1 and cannot be the dot product of two unit vectors.

Final Answer:
The impossible value for \( \mathbf{p} \cdot \mathbf{q} \) is \( \sqrt{3} \).