Question

If $\bar a,\bar b,\bar c$ are position vectors of vertices $A,B,C$ of $\Delta ABC$ . If $\bar r$ is a position vector of a point $P$ such that $\left( {\left| {\bar b - \bar c} \right| + \left| {\bar c - \bar a} \right| + \left| {\bar a - \bar b} \right|} \right)\bar r = \left| {\bar b - \bar c} \right|\bar a + \left| {\bar c - \bar a} \right|\bar b + \left| {\bar a - \bar b} \right|\bar c$ then the point $P$ is
A) Centroid of $\Delta ABC$
B) Orthocentre of $\Delta ABC$
C) Circumcentre of $\Delta ABC$
D) Incentre of $\Delta ABC$

Answer 1

Hint:
First we have given the position vectors of all the vertices so we will use the position vectors to determine the vector representation of each side. Now modulus of the vector will represent the length of the side. We will then rearrange the terms of the given equality and observe what it represents.

Complete step by step solution:
It is given that $\bar a,\bar b,\bar c$ are position vectors of vertices $A, B, C$ of $\Delta ABC$.
Also, it is given that $\bar r$ represents the position vector of point $P$.
Now, the modulus of the difference between the position vectors will represent the length of each side.
We will find the length of each side in terms of the position vector corresponding to the vertices.
First, we will consider the side $AB$.
The length of the side will be given by:
$AB = \left| {\bar b - \bar a} \right|$
Similarly, now we will consider the side $BC$.
The length of the side will be given by:
$BC = \left| {\bar c - \bar b} \right|$
Finally, we will consider the side $CA$.
The length of the side will be given by:
$CA = \left| {\bar a - \bar c} \right|$
Now we have determined all the lengths.
We will consider the given equation as follows:
$\left( {\left| {\bar b - \bar c} \right| + \left| {\bar c - \bar a} \right| + \left| {\bar a - \bar b} \right|} \right)\bar r = \left| {\bar b - \bar c} \right|\bar a + \left| {\bar c - \bar a} \right|\bar b + \left| {\bar a - \bar b} \right|\bar c$
Rearrange the given terms and express the above equation for the position vector $\bar r$ .
$\bar r = \dfrac{{\left| {\bar b - \bar c} \right|\bar a + \left| {\bar c - \bar a} \right|\bar b + \left| {\bar a - \bar b} \right|\bar c}}{{\left( {\left| {\bar b - \bar c} \right| + \left| {\bar c - \bar a} \right| + \left| {\bar a - \bar b} \right|} \right)}}$
Now the denominator is the sum of lengths of the triangle that represents the parameter. Also, the numerator is of the form $ax + by + cz$. Thus, the whole expression represents the incentre of the triangle.

Hence, the correct option is D.

Note:
Note that if you are not able to visualise the process then draw the diagram for better understanding. Also, it is important to observe the given equation and rearrange the terms accordingly. Finally interpret the obtained result to reach the final answer.