The x-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 the two media are given by unit vectors `u_(1) = a hat(i)+b hat(j)` and `u_(2) = c hat(i) +a hat(j)`. Then

Explanation

` tan i=a/b so sin I (a)/(sqrt(a^2 +b^2))`
` and tan r= (C )/(d ) , sin r = (c )/( sqrt(c^2 +d^2))`
` mu_1 sin I = mu_2 sin r`
` ((3)/(2)) ((a)/(sqrt(a^2 +b^2)))=2 ((c )/(sqrt(c^2 +d^2)))`
But as ` a hati + b hatj and c hati + d hatj ` are unit vectors so
` sqrt(a^2 +b^2 ) = sqrt(c^2 +d^2)=1`
hence `3/2 a= 2c , so (a)/(c ) = (4 )/(3) `