Direction cosines $l,m and n$ of given vector $\overrightarrow{v}$,

$l=\cos \left({60}^{\circ }\right)=\frac{1}{2}$

$m=\cos \left({45}^{\circ }\right)=\frac{1}{\sqrt{2}}$

$n=\cos \left(\gamma \right)$

$\overrightarrow{v}=\cos 60°\hat{i}+\cos 45°\hat{j}+\cos \gamma \hat{k}....\left(1\right)$

We know,

$l^2+m^2+n^2=1$

$\Rightarrow \frac{1}{4}+\frac{1}{2}+{\cos }^2\gamma =1 (\gamma \to$Acute$)$

$\Rightarrow \cos \gamma =\frac{1}{2}$

$\Rightarrow \gamma =60°$

Equation of plane is $\frac{1}{2}\left(x-\sqrt{2}\right)+\frac{1}{\sqrt{2}}\left(y+1\right)+\frac{1}{2}\left(z-1\right)=0$

$\Rightarrow \frac{x}{2}+\frac{y}{\sqrt{2}}+\frac{z}{2}=\frac{1}{2}$

$\Rightarrow x+\sqrt{2}y+z=1$

$\left(a,b,c\right)$ lies on it,

$\Rightarrow a+\sqrt{2}b+c=1$