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.