Question

Let \(\vec{a}\) and \(\vec{b}\) be two unit vectors such that their sum is also a unit vector. If \(\vec{u}=\vec{a}+2\vec{b}\) and \(\vec{v}=2\vec{a}+\vec{b}\), then the angle between \(\vec{u}\) and \(\vec{v}\) is

• A. \(0\)
• B. \(\frac{\pi }{6}\)
• C. \(\frac{\pi }{3}\)
• D. \(\frac{\pi }{2}\)

Hint:

Use the condition on the magnitude of the sum to find the dot product of a and b, then compute the angle between u and v.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept
Since \(|\vec{a}|=|\vec{b}|=|\vec{a}+\vec{b}|=1\), squaring the last gives \(1+1+2\vec{a}\cdot\vec{b}=1\), so \(\vec{a}\cdot\vec{b}=-\frac{1}{2}\).

Step 2: Dot product and magnitudes
\(\vec{u}\cdot\vec{v}=2+5\,\vec{a}\cdot\vec{b}+2=4-\frac{5}{2}=\frac{3}{2}\).
\(|\vec{u}|^{2}=1+4+4\vec{a}\cdot\vec{b}=3\) and \(|\vec{v}|^{2}=4+1+4\vec{a}\cdot\vec{b}=3\).

Step 3: Find the angle
\(\cos\theta=\frac{3/2}{\sqrt{3}\sqrt{3}}=\frac{1}{2}\), so \(\theta=\frac{\pi}{3}\).

Final Answer:
The angle is \(\frac{\pi}{3}\). \[ \boxed{\text{(C) }\dfrac{\pi }{3}} \]