JEE PYQ: Vector Algebra - Question ID a34421321612 (JEE Main 2022)

When two unit vectors $\widehat{A}$ and $\widehat{B}$ enclose an angle $\theta$, their sum and difference can be expressed using the following fundamental results from vector algebra:

• Magnitude of the sum: $|\widehat{A} + \widehat{B}| = \sqrt{2 + 2\cos\theta} = 2\cos\left(\frac{\theta}{2}\right)$
• Magnitude of the difference: $|\widehat{A} - \widehat{B}| = \sqrt{2 - 2\cos\theta} = 2\sin\left(\frac{\theta}{2}\right)$

These follow from the identity $|\vec{v}|^2 = \vec{v} \cdot \vec{v}$ and the double-angle formulas for cosine and sine.

Step 1: Express the magnitudes explicitly

Step 2: Form the ratio of the magnitudes

Step 3: Rearrange to match the given options

This exactly matches option B.

Many students confuse the roles of sine and cosine in the magnitudes of the sum and difference. A frequent error is to swap the expressions, leading to incorrect options like A or D. Always derive the magnitudes from first principles using the dot product to avoid this pitfall.