Answer to Question #342844 in Analytic Geometry for guy

Question #342844

Find the position vector of the point in space P that lies on the line AB such that |AP|/|PB|= m/n with m, n is a real number. Find the position vector of P if vector A=(1,2,1), B(3,-1,2), m=3 and n=2


1
Expert's answer
2022-05-22T17:59:27-0400

Let "O" be the origin representing the position vectors "\\overrightarrow{OA}"  and "\\overrightarrow{OB}"  with respect to the two points "A"  and "B" . Let "AB" be divided by a third point "P"  internally in the ratio "m:n."


"\\dfrac{AP}{BP}=\\dfrac{m}{n}=>n\\overrightarrow{AP}=m\\overrightarrow{PB}"

"n(\\overrightarrow{OP}-\\overrightarrow{OA})=m(\\overrightarrow{OB}-\\overrightarrow{OP})"

"\\overrightarrow{OP}=\\dfrac{m\\overrightarrow{OB}+n\\overrightarrow{OA}}{n+m}"

Given


"\\overrightarrow{OA}=\\langle1,2,1 \\rangle, \\overrightarrow{OB}=\\langle3,-1,2 \\rangle"

"m=3, n=2"

"\\overrightarrow{OP}=\\dfrac{3\\langle1,2,1 \\rangle+2\\langle3,-1,2 \\rangle}{2+3}=\\langle1.8,0.8,1.4 \\rangle"

"\\overrightarrow{OP}=\\langle1.8,0.8,1.4 \\rangle"


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