Two particles are located at equal distance from origin. The position vectors of those are represented by $\vec{A}=2 \hat{i}+3 n \hat{j}+2 \hat{k}$ and $\overrightarrow{\mathrm{B}}=2 \hat{\mathrm{i}}-2 \hat{\mathrm{j}}+4 \mathrm{p} \hat{\mathrm{k}}$, respectively. If both the vectors are at right angle to each other, the value of $\mathrm{n}^{-1}$ is $\_\_\_\_$ .

JEE Main · 2025 · 23 Jan 2025 · Shift 1 · Vector

Two particles are located at equal distance from origin. The position vectors of those are represented by

A = 2 \hat{i} + 3 n \hat{j} + 2 \hat{k}

and

B = 2 \hat{i} - 2 \hat{j} + 4 p \hat{k}

, respectively. If both the vectors are at right angle to each other, the value of

n^{- 1}

____

Answer: 3

Two conditions apply to

A = 2 \hat{i} + 3 n \hat{j} + 2 \hat{k}

and

B = 2 \hat{i} - 2 \hat{j} + 4 p \hat{k}

: equal magnitude (both equidistant from the origin) and perpendicularity.

Equal magnitude:

∣ A ∣^{2} = ∣ B ∣^{2}

4 + 9 n^{2} + 4 = 4 + 4 + 16 p^{2} \Rightarrow 9 n^{2} = 16 p^{2} \Rightarrow p = \pm \frac{3 n}{4}

Perpendicularity:

A \cdot B = 0

4 - 6 n + 8 p = 0

Taking

p = \frac{3 n}{4}

4 - 6 n + 6 n = 4 \neq = 0

— no solution.

Taking

p = - \frac{3 n}{4}

4 - 6 n - 6 n = 4 - 12 n = 0 \Rightarrow n = \frac{1}{3}

n^{- 1} = 3

JEE Main · 2025 · 23 Jan 2025 · Shift 1 · Integer answer