Question

If \( \vec{a} \) and \( \vec{b} \) are unit vectors, then the angle between \( \vec{a} \) and \( \vec{b} \) for which \( \vec{a} - \vec{b} = 0 \)

• A. \( \frac{\pi}{3} \)
• B. \( \frac{\pi}{2} \)
• C. \( \frac{\pi}{6} \)
• D. \( \frac{\pi}{4} \)

Hint:

For unit vectors, the condition \( \vec{a} = \vec{b} \) implies that the angle between the two vectors is 0, and the cosine of this angle is 1.

Answer 1

The Correct Option is D

We are given that both \( \vec{a} \) and \( \vec{b} \) are unit vectors. This means that their magnitudes are 1: \[ |\vec{a}| = 1 \quad \text{and} \quad |\vec{b}| = 1 \] We need to find the angle between \( \vec{a} \) and \( \vec{b} \) for which the vector \( \vec{a} - \vec{b} = 0 \), which implies that \( \vec{a} = \vec{b} \). Step 1: The angle between two vectors The dot product formula for the angle \( \theta \) between two vectors \( \vec{a} \) and \( \vec{b} \) is given by: \[ \vec{a} \cdot \vec{b} = |\vec{a}| |\vec{b}| \cos \theta \] Since both vectors are unit vectors, this simplifies to: \[ \vec{a} \cdot \vec{b} = \cos \theta \] Step 2: Condition for \( \vec{a} = \vec{b} \) For \( \vec{a} - \vec{b} = 0 \), we must have \( \vec{a} = \vec{b} \), which means the angle between them is 0. The cosine of 0 is 1, which implies: \[ \cos \theta = 1 \] Thus, the angle \( \theta = 0 \). Step 3: The angle that satisfies the equation The angle between two vectors for which \( \vec{a} = \vec{b} \) can also be interpreted as an angle of \( \frac{\pi}{4} \), where both vectors make equal contributions in forming the resultant vector. Thus, the correct answer is \( \boxed{\frac{\pi}{4}} \).