Question

If \(\vec{a},\vec{b},\vec{c}\) are unit vectors satisfying \(\vec{a}-(\sqrt{3})\vec{b}+\vec{c}=\vec{0}\), then the angle between \(\vec{a}\) and \(\vec{c}\) is...

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

Hint:

Rearrange to a + c = root 3 times b and take the square of the magnitude.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept
All three vectors are unit vectors, so \(|\vec{a}|=|\vec{b}|=|\vec{c}|=1\). Rearrange to \(\vec{a}+\vec{c}=\sqrt{3}\,\vec{b}\).

Step 2: Square the Magnitude
\(|\vec{a}+\vec{c}|^{2}=3|\vec{b}|^{2}=3\).

Step 3: Expand
\(|\vec{a}|^{2}+|\vec{c}|^{2}+2\vec{a}\cdot\vec{c}=3\Rightarrow1+1+2\cos\theta=3\).

Step 4: Solve
\(\cos\theta=\dfrac{1}{2}\), so \(\theta=\dfrac{\pi}{3}\), option (B).

Final Answer:
The angle between a and c is pi/3. \[ \boxed{\text{(B) }\dfrac{\pi}{3}} \]