Question

If two vectors \(\vec{A}\) and \(\vec{B}\) having equal magnitude \(R\) are inclined at an angle \(\theta\), then

• A. \(|\vec{A} - \vec{B}| = \sqrt{2} R \sin \left(\frac{\theta}{2}\right)\)
• B. \(|\vec{A} + \vec{B}| = 2 R \sin \left(\frac{\theta}{2}\right)\)
• C. \(|\vec{A} + \vec{B}| = 2 R \cos \left(\frac{\theta}{2}\right)\)
• D. \(|\vec{A} - \vec{B}| = 2 R \cos \left(\frac{\theta}{2}\right)\)

Answer 1

The Correct Option is C

To address this problem, we must ascertain the resultant magnitude of the vector sum or difference of two vectors, \(\vec{A}\) and \(\vec{B}\). These vectors possess equal magnitudes and are oriented at an angle \(\theta\).

We commence with the formula for vector addition of two vectors \(\vec{A}\) and \(\vec{B}\), assuming they have equal magnitudes, denoted by \(R\):

\(|\vec{A} + \vec{B}| = \sqrt{A^2 + B^2 + 2AB \cos \theta}\)

Given that \(|\vec{A}| = |\vec{B}| = R\), we substitute these values into the aforementioned formula:

\(|\vec{A} + \vec{B}| = \sqrt{R^2 + R^2 + 2 \cdot R \cdot R \cdot \cos \theta}\)

Upon simplification, the expression becomes:

\(|\vec{A} + \vec{B}| = \sqrt{2R^2(1 + \cos \theta)}\)

Employing the trigonometric identity \(1 + \cos \theta = 2 \cos^2 \left(\frac{\theta}{2}\right)\), we perform the substitution to obtain:

\(|\vec{A} + \vec{B}| = \sqrt{2R^2 \cdot 2 \cos^2 \left(\frac{\theta}{2}\right)}\)

This simplifies to:

\(|\vec{A} + \vec{B}| = 2R \cos \left(\frac{\theta}{2}\right)\)

Consequently, the correct result is:

\(|\vec{A} + \vec{B}| = 2 R \cos \left(\frac{\theta}{2}\right)\)

This result aligns with the provided correct option: \(|\vec{A} + \vec{B}| = 2 R \cos \left(\frac{\theta}{2}\right)\).

An examination of the alternative options further substantiates this conclusion:

• \(|\vec{A} - \vec{B}| = \sqrt{2} R \sin \left(\frac{\theta}{2}\right)\): This does not correspond to the derived formula for \(|\vec{A} - \vec{B}|\).
• \(|\vec{A} + \vec{B}| = 2 R \sin \left(\frac{\theta}{2}\right)\): This is inconsistent with the derived formula.
• \(|\vec{A} - \vec{B}| = 2 R \cos \left(\frac{\theta}{2}\right)\): This is incorrect as it represents the addition formula under an erroneous identity.

Therefore, Option 3 is the accurate and valid choice.