A parallel beam of monochromatic light having a wavelength of 663...

The momentum of a single photon is given by $p = \dfrac{h}{\lambda}$.

When a photon strikes a totally reflecting mirror at an angle of incidence $\theta$, the component of its momentum perpendicular to the mirror is $p \cos \theta$. Upon reflection, this perpendicular component reverses its direction, while the parallel component remains unchanged.

The change in momentum of each photon is:
$\Delta p = 2p \cos \theta = \dfrac{2h}{\lambda} \cos \theta$

The force exerted on the mirror is the total change in momentum per second, which is the product of the number of photons striking per second ($n$) and the change in momentum of each photon:
$F = n \Delta p = \dfrac{2nh}{\lambda} \cos \theta$

Substituting the given values ($n = 1.0 \times 10^{19} \text{ s}^{-1}$, $h = 6.63 \times 10^{-34} \text{ J s}$, $\lambda = 663 \times 10^{-9} \text{ m}$, \theta = 60^{}^{{}^{\circ}}):
F = \dfrac{2 \times 1.0 \times 10^{19} \times 6.63 \times 10^{-34}}{663 \times 10^{-9}} \times \cos 60^{}^{{}^{\circ}}

$F = 2 \times 10^{19} \times 10^{-27} \times \dfrac{1}{2}$

$F = 1.0 \times 10^{-8} \text{ N}$