JEE PYQ: Vector Algebra - Question ID 38ce486d407d (JEE Main 2022)

Core Formula & Concept:

To find the magnitude of the component of vector $\vec{A}$ along vector $\vec{B}$, we use the concept of the projection of one vector onto another. The key formula is:

$\text{Component of } \vec{A} \text{ along } \vec{B} = \frac{\vec{A} \cdot \vec{B}}{|\vec{B}|}$

Here:

• $\vec{A} \cdot \vec{B}$ is the dot product of vectors $\vec{A}$ and $\vec{B}$.
• $|\vec{B}|$ is the magnitude of vector $\vec{B}$.

This formula arises because the dot product $\vec{A} \cdot \vec{B} = |\vec{A}||\vec{B}|\cos\theta$, where $\theta$ is the angle between the vectors. Dividing by $|\vec{B}|$ isolates the component of $\vec{A}$ in the direction of $\vec{B}$.

Step-by-Step Derivation:

Given: $\vec{A} = (2\hat{i} + 3\hat{j} - \hat{k})\, \text{m}, \quad \vec{B} = (\hat{i} + 2\hat{j} + 2\hat{k})\, \text{m}$ Step 1: Compute the Dot Product $\vec{A} \cdot \vec{B}$
The dot product is calculated as: $\vec{A} \cdot \vec{B} = A_x B_x + A_y B_y + A_z B_z$ Substitute the components: $\vec{A} \cdot \vec{B} = (2)(1) + (3)(2) + (-1)(2) = 2 + 6 - 2 = 6\, \text{m}^2$ Step 2: Compute the Magnitude of $\vec{B}$
The magnitude of $\vec{B}$ is: $|\vec{B}| = \sqrt{B_x^2 + B_y^2 + B_z^2} = \sqrt{1^2 + 2^2 + 2^2} = \sqrt{1 + 4 + 4} = \sqrt{9} = 3\, \text{m}$ Step 3: Calculate the Component of $\vec{A}$ along $\vec{B}$
Using the formula: $\text{Component of } \vec{A} \text{ along } \vec{B} = \frac{\vec{A} \cdot \vec{B}}{|\vec{B}|} = \frac{6}{3} = 2\, \text{m}$ Final Answer: The magnitude of the component of $\vec{A}$ along $\vec{B}$ is 2 m.

Common Traps & Exam Tip:

1. Confusing Dot Product with Cross Product: Students often mistakenly use the cross product formula instead of the dot product. Remember, the component along a vector requires the dot product, not the cross product.
2. Incorrect Magnitude Calculation: Forgetting to take the square root or misapplying the formula for magnitude (e.g., adding components directly) leads to errors. Always verify the magnitude calculation.
3. Sign Errors in Components: The negative sign in $\vec{A}$'s $\hat{k}$ component is easy to overlook. Double-check all signs when computing the dot product.
4. Unit Mismatch: Ensure all components are in the same units (here, meters) to avoid dimensional inconsistencies in the final answer.

Exam Tip: For projection problems, always write down the formula first and then substitute values systematically. This reduces errors and makes your solution easier to review.