Question

Let \( \vec{x} \) and \( \vec{y} \) be two unit vectors and \( \theta \) is the angle between them. Then \( \vec{x} + \vec{y} \) is a unit vector if

• A. \( \theta = \frac{\pi}{4} \)
• B. \( \theta = \frac{\pi}{3} \)
• C. \( \theta = \frac{\pi}{2} \)
• D. \( \theta = \frac{2\pi}{3} \)

Hint:

To determine when the sum of two unit vectors is a unit vector, use the formula for the magnitude of the sum and solve for \( \theta \).

Answer 1

The Correct Option is A

Step 1: Use the formula for the magnitude of the sum of two vectors.
The magnitude of \( \vec{x} + \vec{y} \) is given by: \[ |\vec{x} + \vec{y}| = \sqrt{|\vec{x}|^2 + |\vec{y}|^2 + 2 |\vec{x}| |\vec{y}| \cos \theta}. \] Since both \( \vec{x} \) and \( \vec{y} \) are unit vectors, \( |\vec{x}| = |\vec{y}| = 1 \), so: \[ |\vec{x} + \vec{y}| = \sqrt{1 + 1 + 2 \cos \theta} = \sqrt{2 + 2 \cos \theta}. \]
Step 2: Set the magnitude equal to 1 for it to be a unit vector.
For \( \vec{x} + \vec{y} \) to be a unit vector, its magnitude must be 1: \[ \sqrt{2 + 2 \cos \theta} = 1. \] Squaring both sides: \[ 2 + 2 \cos \theta = 1 \quad \Rightarrow \quad \cos \theta = -\frac{1}{2}. \]
Step 3: Conclusion.
This equation holds true when \( \theta = \frac{\pi}{4} \), corresponding to option (A).