Question

If \( \vec{a} \) and \( \vec{b} \) are two vectors such that \( |\vec{a}| = 1 \), \( |\vec{b}| = 2 \), and \( \vec{a} \cdot \vec{b} = \sqrt{3} \), then the angle between \( 2\vec{a} \) and \( -\vec{b} \) is:

• A. \( \frac\pi6 \)
• B. \( \frac\pi3 \)
• C. \( \frac5\pi6 \)
• D. \( \frac11\pi6 \)

Hint:

The angle between scaled vectors depends only on the original vectors' angle.

Answer 1

The Correct Option is C

Step 1: Determine the angle between \( \vec{a} \) and \( \vec{b} \). Using the dot product formula: \[ \vec{a} \cdot \vec{b} = |\vec{a}||\vec{b}|\cos\theta \implies \sqrt{3} = (1)(2)\cos\theta \implies \cos\theta = \frac{\sqrt{3}}{2}. \] Therefore, \( \theta = \frac{\pi}{6} \).

Step 2: Calculate the angle between \( 2\vec{a} \) and \( -\vec{b} \). Due to scalar multiplication, the angle is: \[ \pi - \frac{\pi}{6} = \frac{5\pi}{6}. \]
Step 3: Evaluate the provided options. The calculated angle is \( \frac{5\pi}{6} \), which corresponds to option (C).