JEE PYQ: Vector Algebra - Question ID 2ca9d57a4cdb (JEE Main 2022)

In vector algebra, the magnitude of the sum and difference of two vectors $\overrightarrow{A}$ and $\overrightarrow{B}$ can be expressed using the dot product. The key formulas are:

• Magnitude of sum: $|\overrightarrow{A} + \overrightarrow{B}| = \sqrt{|\overrightarrow{A}|^2 + |\overrightarrow{B}|^2 + 2 \overrightarrow{A} \cdot \overrightarrow{B}}$
• Magnitude of difference: $|\overrightarrow{A} - \overrightarrow{B}| = \sqrt{|\overrightarrow{A}|^2 + |\overrightarrow{B}|^2 - 2 \overrightarrow{A} \cdot \overrightarrow{B}}$

Since the magnitudes of $\overrightarrow{A}$ and $\overrightarrow{B}$ are equal, let $|\overrightarrow{A}| = |\overrightarrow{B}| = A$. The dot product $\overrightarrow{A} \cdot \overrightarrow{B}$ can also be written as $A^2 \cos \theta$, where $\theta$ is the angle between the two vectors.

Given: $|\overrightarrow{A} + \overrightarrow{B}| = 2 |\overrightarrow{A} - \overrightarrow{B}|$

Step 1: Express the magnitudes using the formulas above.

Step 2: Substitute these into the given condition.

Step 3: Square both sides to eliminate the square roots.

Step 4: Simplify by dividing both sides by $2A^2$ (since $A \neq 0$).

Step 5: Expand and solve for $\cos \theta$.

Step 6: Determine the angle $\theta$.

Thus, the correct answer is option C.

Students often make the following mistakes:

• Forgetting to square the magnitudes when equating the expressions, leading to incorrect simplification.
• Misapplying the formula for the magnitude of the sum or difference of vectors, especially the sign of the dot product term.
• Confusing $\sin^{-1}$ with $\cos^{-1}$ in the final step, as both trigonometric functions appear in the options.

Exam Tip: Always verify the derived trigonometric function matches the options. Here, since we derived $\cos \theta = \frac{3}{5}$, the correct answer must involve $\cos^{-1}$.