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  • Find the position vector of a point R which divides the line joining two points P and Q whose position vectors are externally in the ratio 1 : 2. Also, show that P is the mid point of the line segment RQ.

Q9   Find the position vector of a point R which divides the line joining two points P and Q whose position vectors are ( 2 \vec a + \vec b ) \: \:and \: \: ( \vec a - 3 \vec b ) externally in the ratio 1: 2. Also, show that P is the mid point of the line segment RQ.

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Given, two vectors \vec P=( 2 \vec a + \vec b ) \: \:and \: \:\vec Q= ( \vec a - 3 \vec b )

the point  R which divides line segment PQ in ratio 1:2 is given by 

=\frac{2(2\vec a +\vec b)-(\vec a-3\vec b)}{2-1}=4\vec a +2\vec b -\vec a+3\vec b=3\vec a+5\vec b

Hence position vector of R is 3\vec a+5\vec b.

Now, Position vector of the midpoint of RQ

=\frac{( 3\vec a + 5\vec b + \vec a - 3 \vec b )}{2}=2\vec a+\vec b

which is the position vector of Point P . Hence, P is the mid-point of RQ

Posted by

Pankaj Sanodiya

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