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Question
Let `A (bara)` and `B (barb)` are any two points in the space and `"R"(bar"r")` be a point on the line segment AB dividing it internally in the ratio m : n, then prove that `bar r = (mbarb + nbara)/(m + n) `
Solution
R is a point on the line segment AB(A – R – B) and `bar("AR")` and `bar("RB")` are in the same direction.
Point R divides AB internally in the ratio m : n
∴ `("AR")/("RB") = m/n`
∴ n(AR) = m(RB)
As `n(bar("AR"))` and `m(bar("RB"))` have same direction and magnitude,
`n(bar("AR")) = m(bar("RB"))`
∴ `n(bar("OR") - bar("OA")) = m(bar("OB") - bar("OR"))`
∴ `n(vecr - veca) = m(vecb - vecr)`
∴ `nvecr - nveca = mvecb - mvecr`
∴ `mvecr + nvecr = mvecb + nveca`
∴ `(m + n)vecr = mvecb + nveca`
∴ `vecr = (mvecb + nveca)/(m + n)`
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