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

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 $\_\_\_\_$ ．\

Professor PC Thomas Classes · Answer

(3) :From equation (i) and (ii), $$ \begin{aligned} & |\vec{A}|=|\vec{B}| \Rightarrow \sqrt{8+9 n^{2}}=\sqrt{8+16 p^{2}} \ \Rightarrow \quad & 3 n= \pm 4 p \end{aligned} $$ As both the vectors are right angle to each other, so dot product $\vec{A}$ and $\vec{B}$ is,\ $\vec{A} \cdot \vec{B}=4-6 n+8 p=0 \Rightarrow 4-6 n+8 p=0$\ $\Rightarrow 4+8 p=6 n$\ Using $p=\frac{-3 n}{4} ; 4+8\left(\frac{-3 n}{4}ight)=6 n$\ $\Rightarrow \quad 4-6 n=6 n \Rightarrow 4=12 n \Rightarrow n=\frac{1}{3}$ or $n^{-1}=3$\