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…

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