A sailor in a boat, which is going due east with a speed of `8m//s` observes that a submarine is heading towards north at a speed of `12m//s` and sinking at a rate of `2 m//s`.The commander of submarine observes a helicopter ascending at a rate of `5 m//s` and heading towards west with `4 m//s`.Find the speed of the helicopter with respect to boat.

Explanation

Taking east, north and vertically upward directions along `+x`, `+y` and `+z` axes respectively.
For boat `vecV_(B)=8hati`
Relative velocity of submarine w.r.t. boat `vecV_(S//B)=12hatj-2hatk`
`:. vecV_(S)-vecV_(B)=12hatj-2hatk rArr vecV_(S)=8hati+12hatj-2hatk`
Relative velocity of helicopter w.r.t. submarine
`vecV_(H//S)=vecV_(H)-vecV_(S)`
`5hatk-4hati=vecV_(H)-(8hati+12hatj-2hatk)`
`vecV_(H)=+4hati+12hatj+3hatk`
`|vecV_(H)|=sqrt(4^(2)+12^(2)+3^(2))=13m//s`
`vecV_(H//B)=vecV_(H)-vecV_(B)=(4hati+12hatj+3hatk)-8hati= -4hati+12hatj+3hatk`
`|vecV_(H//B)|=13m//s`