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Question 189 of 215

Q.If A(aˉ)A(\bar{a}) and B(bˉ)B(\bar{b}) are any two points in the space and R(rˉ)R(\bar{r}) be a point on the line segment AB dividing it internally in the ratio m:nm:n, then prove that rˉ=mbˉ+naˉm+n\bar{r} = \dfrac{m\bar{b} + n\bar{a}}{m+n}

Maharashtra MsbshseMaharashtra HSC (MSBSHSE) Board 2019Subjective· 3mImportance★★★★★
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Use AR⃗:RB⃗=m:n\vec{AR}:\vec{RB}=m:n along the same line, and express both in terms of position vectors.

Let OO be the origin, with position vectors OA⃗=aˉ\vec{OA}=\bar a, OB⃗=bˉ\vec{OB}=\bar b, OR⃗=rˉ\vec{OR}=\bar r, where RR lies on segment ABAB dividing it internally so that AR:RB=m:nAR:RB=m:n.

Since RR lies between AA and BB on the same line, AR⃗\vec{AR} and RB⃗\vec{RB} point in the same direction, and

n⋅AR⃗=m⋅RB⃗n\cdot\vec{AR} = m\cdot\vec{RB}

Now AR⃗=rˉ−aˉ\vec{AR} = \bar r - \bar a and RB⃗=bˉ−rˉ\vec{RB} = \bar b - \bar r. Substituting:

n(rˉ−aˉ)=m(bˉ−rˉ)n(\bar r-\bar a) = m(\bar b-\bar r)

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