Question

If three vectors have equal magnitude i.e. $A = B = C$, then the angle between $\vec{A$ and $\vec{C}$ is $\alpha$. If $\vec{A} + \vec{B} + \vec{C} = 0$, then the angle between $\vec{A}$ and $\vec{C}$ is $\beta$, then $\dfrac{\alpha}{\beta}$ is}

• A. $\dfrac{2}{3}$
• B. $\dfrac{2}{1}$
• C. $\dfrac{1}{2}$
• D. $\dfrac{3}{2}$

Hint:

Three equal vectors in equilibrium always subtend angles of $120^\circ$ with each other.

Answer 1

The Correct Option is C

Step 1: Case of three equal vectors.
When three vectors of equal magnitude form a closed triangle, the angle between any two vectors is $120^\circ$.
Hence, $\alpha = 120^\circ$.

Step 2: Given condition $\vec{A + \vec{B} + \vec{C} = 0$.}
This represents equilibrium of three equal vectors.
Therefore, the angle between each pair of vectors is $120^\circ$.
Hence, $\beta = 60^\circ$.

Step 3: Ratio calculation.
\[ \frac{\alpha}{\beta} = \frac{120^\circ}{60^\circ} = \frac{1}{2} \]

Step 4: Conclusion.
The required ratio is $\dfrac{1}{2}$.