The Y-Z plane separates two media A and B of refractive indices ${\mu }_1=1.5$ and ${\mu }_2=2$.  A ray of light travels from A to B. Its directions in two media are given by unit vectors $\overrightarrow{u_1}=a\hat{i} + b\hat{j} and \overrightarrow{u_2}= c\hat{i} + d \hat{j}$ . Then

      

as the light rays are falling on the yz plane, X axis is perpendicular to the plane.

For angle of incidence with X axis

tan i = $\frac{b}{a}$ and 

$\sin i=\frac{b}{\sqrt{a^2+b^2}}=b$

$similarly, \sin r= \frac{d}{\sqrt{c^2+d^2}} =d$

$$\begin{gathered} as U_1 and U_{2 }are unit vectors, \\ \sqrt{a^2+b^2}=\sqrt{c^2+d^2}=1 \end{gathered}$$

$$\begin{gathered} according to Snell\text{'}s law, \\ {\mu }_1\sin i ={\mu }_2\sin r \end{gathered}$$

$$\begin{gathered} \Rightarrow \frac{3}{2}b=2d \\ \Rightarrow \frac{b}{d}=\frac{4}{3} \end{gathered}$$