Two particles are located at equal distance from origin. The position vectors of those are represented by and , respectively. If both the vectors are at right angle to each other, the value of is __________. [2025]

Question

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 $\vec{B} = 2 \hat{i} - 2 \hat{j} + 4 p \hat{k}$ , respectively. If both the vectors are at right angle to each other, the value of $n^{- 1}$ is __________. [2025]

Smart Achievers · Accepted Answer

(3)

$\vec{A} \cdot \vec{B} = 0 and$ $| \vec{A} |$ $\vec{B} = 2 \hat{i} - 2 \hat{j} + 4 p \hat{k}$ $| \vec{B} |$

$\Rightarrow$ 4 – 6n + 8p = 0

3n – 4p = 2 ... (i)

Also $4 + 9 n^{2} + 4 = 4 + 4 + 16 p^{2}$

$3 n = \pm 4 p$ ... (ii)

$\pm 4 p - 4 p = 2$

Taking –ve sign

–8p = 2

$p = - \frac{1}{4}, 3 n + 1 = 2 \Rightarrow n = \frac{1}{3}$

Solution

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Solution

Q. Two particles are located at equal distance from origin. The position vectors of those are represented by A→=2i^+3nj^+2k^ and B→=2i^–2j^+4pk^, respectively. If both the vectors are at right angle to each other, the value of n–1 is __________. [2025]

Ans. (3) A→·B→=0 and |A→|=|B→| ⇒ 4 – 6n + 8p = 0 3n – 4p = 2 ... (i) Also 4+9n2+4=4+4+16p2 3n=±4p ... (ii) ±4p–4p=2 Taking –ve sign –8p = 2 p=–14, 3n+1=2 ⇒ n=13

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