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Question

Given the vertices of triangle by position vectors ^i+^j+^k,^i+^k and ^j+^k the centroid and Incentre of the triangle will be given by

A
2^i+2^j+3^k3,(2+1)^i+(2+1)^j+(2+2)^k2+2
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B
2^i+2^j+3^k3,(2+1)^i+(2+2)^j+(2+1)^k2+2
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C
2^i+2^j+3^k3,(2+1)^i+(2+2)^j+(2+2)^k2+2
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D
2^i+2^j+3^k3,(2+2)^i+(2+1)^j+(2+2)^k2+2
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Solution

The correct option is A 2^i+2^j+3^k3,(2+1)^i+(2+1)^j+(2+2)^k2+2
Given position vecto ¯p1,¯p2 and ¯p3. We know centroid is given by G=p1+p2+p33
I=|¯p2¯p3|¯p1+|¯p1¯p3|¯p2+|¯p1¯p2|¯p3|¯p2¯p3|+|¯p2¯p1|+|¯p3¯p1|
Therefor, G=(^i+^j+^k)+(^i+^k)+(^j+^k)3
=2^i+2^j+3^k3
|¯p2¯p3|=|^i+^k^j^k|=|^i^l|=2|¯p2¯p1|=|^i+^k^i^j^k|=|^j|=1|¯p3¯p1|=|^j+^k^i^j^k|=|^i|=1
I=2(^i+^j+^k)+1(^i+^k)+(^j+^k)2+1+1
=(2+1)^i+(2+1)^j+(2+2)^k2+2
Correct option is A.

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