1.

The position vectors of the vertices A,B,C of DeltaABC are hati-hatj-3hatk,2hati+hatj-2hatk and -5hati+2hatj-6hatk respectively. The length of the bisector AD of the angle /_BAC where D is on the line segment BC is

Answer»

`15/2`
`3/4sqrt(10)`
`4/3sqrt(10)`
None of these

Solution :We have
`VEC(AB)=hati+2hatj+hatk, vec(AC)-6hati+3hatj-3hatk`
`implies|vec(AB)|=sqrt(6)` and `|vec(AC)|=3sqrt(6)`
Clearly, point D divides BC in the ratio `AB:AC` i.e. `1:3`.
`:.` Position vector ofDis
`((-5hati+2hatj-6hatk)+3(2hati+hatj-2hatk))/(1+3)`
`implies` Position vector of D is `=1/4(hati+5hatj-12hatk)`
`:.vec(AD)=1/4(hati+5hatj-12hatk)-(hati-hatj-3hatk)`
`vec(AD)=3/4(-hati+3hatj)`
`implies|vec(AD)|=3/4sqrt(10)`


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