Find the distance from the origin to the plane $\overrightarrow{r} \cdot \left(2\hat{i} - \hat{j} + 5 \hat{k}\right) = 7$
Vector equation of given plane is: $\;$ $\overrightarrow{r} \cdot \left(2\hat{i} - \hat{j} + 5 \hat{k}\right) = 7$
$\therefore$ $\;$ The cartesian equation of the given plane is: $\;$ $2x - y + 5z = 7$ $\;\;\; \cdots \; (1)$
Equation $(1)$ is of the form $\;$ $ax + by + cz + d = 0$
where $\;$ $a = 2, \; b = -1, \; c = 5, \; d = -7$
Now, distance from the origin to the plane given by equation $(1)$ is
$\begin{aligned}
\left|\dfrac{d}{\sqrt{a^2 + b^2 + c^2}}\right| & = \left|\dfrac{-7}{\sqrt{\left(2\right)^2 + \left(-1\right)^2 + \left(5\right)^2}}\right| \\\\
& = \left|\dfrac{-7}{\sqrt{4 + 1 + 25}}\right| \\\\
& = \dfrac{7}{30} \; units
\end{aligned}$