direction cosines of reflected ray

Question

In the above question how they have written the value cos . I am not able to understand that , please help .

Answer 1

The source is on OX, so it is a constant (say $x$) multiplied by $\hat{i}$. The normal to the mirror is $\hat{i}-\hat{j}+\hat{k}$. The cosine of the angle between OX and the normal is given by $$\cos\theta=\frac{\hat{i}\cdot(\hat{i}-\hat{j}+\hat{k})}{|\hat{i}||\hat{i}-\hat{j}+\hat{k}|}=\frac{1}{1\sqrt{1^2+(-1)^2+1^2}}=\frac{1}{\sqrt{3}}$$ I used $\hat{i}\hat{i}=\hat{j}\hat{j}=\hat{k}\hat{k}=1$ and $\hat{i}\hat{j}=\hat{i}\hat{k}=0$