Vectors with position vector \(A= \quad 2 \hat{i}+3 n \hat{j}+2 \hat{k} \quad B=\) \(2 \hat{i}-2 \hat{j}+4p \hat{k}\), s…

\(\begin{aligned}
&\begin{aligned}
& \vec{A} \cdot \vec{B}=0 \\
& (2 \hat{i}+3 n \hat{\jmath}+2 \hat{k}) \cdot(2 \hat{i}-2 \hat{\jmath}+4 p \hat{k})=0 \\
& 4-6 n+8 p=0 \\
& 8 p-6 n=-4 \\
& \quad 3 n - 4 p=2 ....\text { (1) } \\
& \text { Given equidistance from origin}
\end{aligned}\\
&\begin{gathered}
|\vec{A}|=|\vec{B}| \\
|\vec{A}|^2=|\vec{B}|^2 \\
4+9 n^2+4=4+4+16 p^2 \\
9 n^2=16 p^2 \\
3 n= \pm 4 p \\
3 n=-4 p \quad or \quad 3 n=4 p \\
8 p=-2 \\
p=-1 / 4 \\
4\left(-\frac{1}{4}\right)-3 n=-2 \\
1-3 n=-2 \\
3 n=1 \\
n=\frac{1}{3} \\
3 n+4 p=3\left(\frac{1}{3}\right)+4\left(-\frac{1}{4}\right) \\
=0
\end{gathered}
\end{aligned}\)0