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

Understanding the dot product of unit vectors.
The dot product \( \mathbf{p} \cdot \mathbf{q} \) of two unit vectors \( \mathbf{p} \) and \( \mathbf{q} \) is given by: \[ \mathbf{p} \cdot \mathbf{q} = \cos \theta \] where \( \theta \) is the angle between the two vectors. Since the cosine of an angle must lie between \( -1 \) and \( 1 \), the value of \( \mathbf{p} \cdot \mathbf{q} \) must be between \( -1 \) and \( 1 \) inclusive. Therefore, \( \sqrt{3} \) is not a possible value for \( \mathbf{p} \cdot \mathbf{q} \), as it exceeds the maximum possible value of 1.