The rain is falling vertically downward with velocity `6m//s` and a man is moving horizontally with velocity `8m//s`. Find the velocity of rain with respect to the man.

Explanation

`v_(r//g)=6m//s, v_(m//g)=8m//s`
`vec(v)_(r//m)=vec(v)_(r//g)-vec(v)_(m//g)=vec(v)_(r//g)+(-vec(v)_(m//g))`
`rarr (v)_(m//g)=8m//s`
`darr (v)_(r//g)=6m//s`
`v_(r//m)=sqrt((6)^(2)+(8)^(2)) =10m//s`
`tan alpha=6/8=3/4 implies alpha=37^(@)`
`tan beta=8/6=4/3 implies beta=53^(@)`
Hence, the velocity of rain with respect to the man is `10 m//s` at an angle `37^(@)` with horizontal or at an angle `53^(@)` with vertical.
OR
`(v)_(r//g)=-6hatk,vec(v)_(m//g)=8hati`
`vec(v)_(r//m)=vec(v)_(r//g)-bar(v)_(m//g)=-6hatk-8hati`
`|vec(v)_(r//m)|=(v)_(r//m)=sqrt((-6)^(2)+(-8)^(2))=10 m//s`