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A(1,-1,-3), B(2, 1,-2) & C(-5, 2,-6) are...

`A(1,-1,-3), B(2, 1,-2) & C(-5, 2,-6)` are the position vectors of the vertices of a triangle ABC. The length of the bisector of its internal angle at A is :

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The correct Answer is:
3
We have, `AB=hati+2hatj+hatk,AC=-6hati+3hatj-3hatk`
`implies|AB|=sqrt(6) and |AC|=3sqrt(6)`
Clearly, point D divides BC in the ratio AB:AC, i.e., 1:3
`therefore`Position vector of `D=((-5hati+2hatj-6hatk)+3(2hati+hatj-2hatk))/(1+3)`
`=(1)/(4)(hati+5hat9-12hatk)`
`thereforeAD=(1)/(4)(hati+5hatj-12hatk)-(hati-hatj-3hatk)=(3)/(4)(-2hati+3hatj)`
`|AD|=AD=(3)/(4)sqrt(10)`
`therefore lamda=3`.
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