Two thin slabs of refractive indices `mu_(1)` and `mu_(2)` are placed parallel to each other in the x-z plane. If the direction of propagation of a ray in the two media are along the vectors `vec(r)_(1) = a hati +b hatj` and `vec(r)_(2) = c hati +d hatj` then we have:

The xz plane separates two media A and B with refractive indices mu_(1) & mu_(2) respectively. A ray of light travels from A to B . Its directions in the two media are given by the unit vectors, vec(r)_(A)=a hat i+ b hat j & vec(r)_(B)= alpha hat i + beta hat j respectively where hat i & hat j are unit vectors in the x & y directions. Then :

The xz plane separates two media A and B with refractive indices mu_(1) & mu_(2) respectively. A ray of light travels from A to B . Its directions in the two media are given by the unut vectors, vec(r)_(A)=a hat i+ b hat j & vec(r)_(B) alpha hat i + beta hat j respectively where hat i & hat j are unit vectors in the x & y directions. Then :

The xz plane separates two media A and B with refractive indices mu_(1) & mu_(2) respectively. A ray of light travels from A to B . Its directions in the two media are given by the unut vectors, vec(r)_(A)=a hat i+ b hat j & vec(r)_(B) alpha hat i + beta hat j respectively where hat i & hat j are unit vectors in the x & y directions. Then :

Prove that the vectors vec(A) = hati +2 hatj +3hatk and vec(B) = 2hati - hatj are perpendicular to each other.

Prove that the vectors vec(A) = 4 hati +3hatj +hatk and vec(B) = 12 hati + 9 hatj +3 hatk are parallel to each other.

Given two vectors veca= -hati +hatj + 2hatk and vecb =- hati - 2 hatj -hatk find |vec a xx vec b|