The vectors `vec(AB)=3hati+4hatk and vec(AC)=5hati-2hatj+4hatk` are the sides of a triangle ABC. The length of the median through A is (A) `sqrt(72)` (B) `sqrt(33)` (C) `sqrt(2880` (D) `sqrt(18)`

Options

A. `sqrt(18)`
B. `sqrt(72)`
C. `sqrt(33)`
D. `sqrt(45)`

Answer

C

Explanation

We know that, the sum of three vectors of a triangle is zero.

`thereforeAB+BC+CA=0`
`implies BC=AC-AB` [`becauseAC=-CA`]
`implies AB=(AC-AB)/(2)" "[because" M is a mid-point of BC"]`
also, `AB+BM+MA=0` [by properties of a triangle]
`impliesAB+(AC-AB)/(2)=AM" "[becauseAM=-MA]`
`implies AM=(AB+AC)/(2)`
`=(3hati+4hatj+5hati-2hatj+4hatk)/(2)`
`=4hati-hatj+4hatk`
`implies|AM|=sqrt(4^(2)+1^(2)+4^(2))=sqrt(33)`.