If $\mathrm{p}\overline{\mathrm{i}}+3\overline{\mathrm{j}}+4\overline{\mathrm{k}}\text{ and }\sqrt{\mathrm{q}}\overline{\mathrm{i}}+5\overline{\mathrm{k}}$ are two vectors where p,q $\ge$ 0 are two scalars. Then the lengths of the vectors are equal for

Given $|\mathrm{p}\overline{\mathrm{i}}+3\overline{\mathrm{j}}+4\overline{\mathrm{k}}|=|\sqrt{\mathrm{q}}\overline{\mathrm{i}}+5\overline{\mathrm{k}}|$
$\Rightarrow \sqrt{{\mathrm{p}}^2+9+16}=\sqrt{\mathrm{q}+25}\Rightarrow {\mathrm{p}}^2+25=\mathrm{q}+25\Rightarrow {\mathrm{p}}^2=\mathrm{q}$
$(\mathrm{p},\mathrm{q}\ge 0)$.Hence there are infinite values of p,q satisfying ${\mathrm{p}}^2=\mathrm{q}(\mathrm{p},\mathrm{q}\ge 0)$