Question

Find the coordinates of the point $P$ which is $\frac{1}{2}$ of the way from $A(-1,-2,5)$ to $B(-2,-2,1)$

Answer 1

Given:$AP=\frac{1}{2}AB$

$\Rightarrow 2AP=AB$

$\Rightarrow 2AP=AP+PB$

$\Rightarrow 2AP-AP=PB$

$\Rightarrow AP=PB$

$\Rightarrow \frac{AP}{PB}=1$

$\Rightarrow P$ divides $AB$ internally in the ratio $1:1$

$\therefore P=(\frac{1\times -2+1\times -1}{1+1},\frac{1\times -2+1\times -2}{1+1},\frac{1\times 1+1\times 5}{1+1})$

$=(\frac{-2-1}{2},\frac{-2-2}{2},\frac{1+5}{2})$

$P=(\frac{-3}{2},\frac{-4}{2},\frac{6}{2})$

$P=(\frac{-3}{2},-2,3)$