Question

Find the position vector of point R which divides the line joining two points P and Q whose position vector is (2$\vec a$+$\vec b$)and($\vec a$-3$\vec b$)externally in the ratio 1:2. Also, show that P is the midpoint of the line segment RQ.

Answer 1

It is given that $\vec{OP}$=2$\vec a$+$\vec b$, $\vec {OQ}$=$\vec a$-3$\vec b$.
It is given that point R divides a line segment joining two points P and Q externally in the ratio 1:2.Then, on using the section formula, we get:
$\vec{OR}$=2(2$\vec a$+$\vec b$)-($\vec a$-3$\vec b$)/2-1

=$\frac{4\vec a+2 \vec b - \vec a+3\vec b }{1}=3\vec a+5\vec b$
Therefore, the position vector of point R is 3$\vec a$+5$\vec b$
Position vector of the mid-point of RQ=$\vec{OQ}$+$\frac{\vec{OR}}{2}$
=$\frac{(\vec a-3\vec b)+(3\vec a+5\vec b)}{2}$
=$2\vec a+\vec b$
=$\vec{OP}$
Hence, P is the mid-point of the line segment RQ.