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 $1:1$

By Section Formula

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

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

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

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