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 OO be the origin representing the position vectors OA\overrightarrow{OA}  and OB\overrightarrow{OB}  with respect to the two points AA  and BB . Let ABAB be divided by a third point PP  internally in the ratio m:n.m:n.


APBP=mn=>nAP=mPB\dfrac{AP}{BP}=\dfrac{m}{n}=>n\overrightarrow{AP}=m\overrightarrow{PB}

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

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

Given


OA=1,2,1,OB=3,1,2\overrightarrow{OA}=\langle1,2,1 \rangle, \overrightarrow{OB}=\langle3,-1,2 \rangle

m=3,n=2m=3, n=2

OP=31,2,1+23,1,22+3=1.8,0.8,1.4\overrightarrow{OP}=\dfrac{3\langle1,2,1 \rangle+2\langle3,-1,2 \rangle}{2+3}=\langle1.8,0.8,1.4 \rangle

OP=1.8,0.8,1.4\overrightarrow{OP}=\langle1.8,0.8,1.4 \rangle


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