Given : Frequency $$f = {3 \over {2\pi }} \times {10^{12}}$$ Hz; direction $$ = {{\widehat i + \widehat j} \over {\sqrt 2 }}$$.

The beam is polarised along $${\widehat k}$$ direction.

Direction of magnetic field $$\overrightarrow B = \left( {{{\widehat i + \widehat j} \over {\sqrt 2 }} \times \widehat k = {{ - \widehat j} \over {\sqrt 2 }} + {{\widehat i} \over {\sqrt 2 }}} \right)$$

Therefore, direction of $$\overrightarrow B = \left( {{{\widehat i - \widehat j} \over {\sqrt 2 }}} \right)$$

Now from wave equation, we have

$$B = {B_0}\cos (kr - \omega t)$$

$$\omega = 2\pi f = 2\pi \times {3 \over {2\pi }} \times {10^{12}}Hz = 3 \times {10^{12}}Hz$$

Now, $$k = {{\widehat i + \widehat j} \over {\sqrt 2 }}$$;

$${B_0} = {{{E_0}} \over c}\left( {{{\widehat i - \widehat j} \over {\sqrt 2 }}} \right)$$

Therefore,

$$B = {{{E_0}} \over c}\left( {{{\widehat i - \widehat j} \over {\sqrt 2 }}} \right)\cos \left[ {{{10}^4}\left( {{{\widehat i + \widehat j} \over {\sqrt 2 }}} \right)\,.\,r - (3 \times {{10}^{12}})t} \right]$$