Given that 

Direction ratios of $\overrightarrow{OP}$ are $\left(1, -2, -2\right)$

Distance between origin and point $P$ is $3 = r.$

Direction cosines of $\overrightarrow{OP}$ are $\left(l, m, n\right) = \left(\frac{1}{3}, \frac{-2}{3},\frac{-2}{3}\right)$

Coordinates of point $P = \left(x, y, z\right) = \left(lr, mr, nr\right) = \left(\frac{1}{3}\times 3, \frac{-2}{3}\times 3, \frac{-2}{3}\times 3\right)= \left(1, -2, -2\right)$