Answer to Question #342844 in Analytic Geometry for guy

Question

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

Answer 1

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

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

$$m=3, n=2$$

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

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