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.
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.
Solution and Explanation
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.
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Concepts Used:
Vector Algebra
A vector is an object which has both magnitudes and direction. It is usually represented by an arrow which shows the direction(→) and its length shows the magnitude. The arrow which indicates the vector has an arrowhead and its opposite end is the tail. It is denoted as
The magnitude of the vector is represented as |V|. Two vectors are said to be equal if they have equal magnitudes and equal direction.
Vector Algebra Operations:
Arithmetic operations such as addition, subtraction, multiplication on vectors. However, in the case of multiplication, vectors have two terminologies, such as dot product and cross product.