JEE PYQ: Vector Algebra - Question ID 5db0c1b33026 (JEE Main 2023)

In vector algebra, two vectors are said to be perpendicular (or orthogonal) if and only if their dot product is zero. The dot product of two vectors \(\vec{A} = A_x \hat{i} + A_y \hat{j} + A_z \hat{k}\) and \(\vec{B} = B_x \hat{i} + B_y \hat{j} + B_z \hat{k}\) is defined as: $\vec{A} \cdot \vec{B} = A_x B_x + A_y B_y + A_z B_z$ If \(\vec{A} \cdot \vec{B} = 0\), then \(\vec{A}\) and \(\vec{B}\) are perpendicular.

In this problem, we are given two vectors \(\vec{P}\) and \(\vec{Q}\) and told they are perpendicular. We will use the dot product condition to find the value of \(m\).

Given vectors: $\vec{P} = \hat{i} + 2m \hat{j} + m \hat{k}$ $\vec{Q} = 4 \hat{i} - 2 \hat{j} + m \hat{k}$

Since \(\vec{P}\) and \(\vec{Q}\) are perpendicular: $\vec{P} \cdot \vec{Q} = 0$

Compute the dot product: $\vec{P} \cdot \vec{Q} = (1)(4) + (2m)(-2) + (m)(m) = 0$ Simplify: $4 - 4m + m^2 = 0$

Rearrange the equation: $m^2 - 4m + 4 = 0$ This is a quadratic equation in \(m\).

Solve the quadratic equation: $m = \frac{4 \pm \sqrt{(-4)^2 - 4 \cdot 1 \cdot 4}}{2} = \frac{4 \pm \sqrt{16 - 16}}{2} = \frac{4 \pm 0}{2} = 2$ Thus, \(m = 2\).

The value of \(m\) that satisfies the condition is \(2\), which corresponds to option D.

1. Sign Errors in Dot Product: Students often make mistakes in the sign while computing the dot product, especially when dealing with negative components. For example, in this problem, the \(j\)-component of \(\vec{Q}\) is \(-2\), and multiplying it with \(2m\) gives \(-4m\), not \(+4m\).

2. Forgetting the Dot Product Condition: Some students confuse the condition for perpendicular vectors with that for parallel vectors. Remember: perpendicular vectors have a dot product of zero, while parallel vectors are scalar multiples of each other.

3. Quadratic Equation Mistakes: While solving the quadratic equation, students may incorrectly compute the discriminant or misapply the quadratic formula. Always double-check the discriminant calculation.

Exam Tip: When solving such problems, always write down the dot product explicitly and verify each term before simplifying. This minimizes errors and ensures accuracy.