Question

Point P partitions directed line segment AB in a 3:1 ratio. Find the coordinates of P, given A (3, 5) and B (-5,1)

Answer 1

Hi,

Explanation:

`A=(3,5)\ \Longrightarrow\ \overrightarrow{OA}=3*\vec{i}+5*\vec{j}\\B=(-5,1)\ \Longrightarrow\ \overrightarrow{OB}=-5*\vec{i}+1*\vec{j}\\\overrightarrow{AP}=3*\overrightarrow{PB}\Longrightarrow\ \overrightarrow{AB}=\overrightarrow{AP}+\overrightarrow{PB}=4*\overrightarrow{PB}\\\overrightarrow{AB}=(-5-3)*\vec{i}+(1-5)*\vec{j}\\\\`

`\overrightarrow{BP}= \frac{\overrightarrow{AB}}{4} =2*\vec{i}+\vec{j}\\\\\overrightarrow{OP}=\overrightarrow{OA}+\overrightarrow{AP}\\\overrightarrow{OP}=\overrightarrow{OB}+\overrightarrow{BP}\\\\2*\overrightarrow{OP}=\overrightarrow{OA}+\overrightarrow{OB}-2*\overrightarrow{BP}\\=3*\vec{i}+5*\vec{j}+(-5)*\vec{i}+\vec{j}-2*(2*\vec{i}+\vec{j})\\=-6*\vec{i}+4*\vec{j}\\\overrightarrow{OP}=-3*\vec{i}+2*\vec{j}\\P=(-3,2)\\`