Find the length of the perpendicular from the origin to the plane $\;$ $\overrightarrow{r} \cdot \left(3 \hat{i} + 4 \hat{j} + 12 \hat{k}\right) = 26$
Equation of the given plane is $\;$ $\overrightarrow{r} \cdot \left(3 \hat{i} + 4 \hat{j} + 12 \hat{k}\right) = 26$
which is of the form $\;$ $\overrightarrow{r} \cdot \overrightarrow{n} = d$
where $\;$ $\overrightarrow{n} = 3 \hat{i} + 4 \hat{j} + 12 \hat{k}$, $\;$ $d = 26 > 0$
Length of the perpendicular from the origin to the plane is
$\dfrac{d}{\left|\overrightarrow{n}\right|} = \dfrac{26}{\sqrt{\left(3\right)^2 + \left(4\right)^2 + \left(12\right)^2}} = \dfrac{26}{13} = 2$ units