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Question

Let ABC be a triangle whose circumcentre is at P. If the position vectors of A,B,C and P are a,b,c and a+b+c4 respectively, then the position vector of the orthocentre of this trianlge, is:

A
a+b+c
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B
0
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C
(a+b+c2)
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D
a+b+c2
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Solution

The correct option is D a+b+c2
Position vector of centroid of ABC=a+b+c3
We know that orthocentre, centroid and circumcentre of any triangle are collinear and the centroid divides the line-segment joining orthocentre and circumcentre in the ratio 2:1.
a+b+c3=2(a+b+c4)+1(P.V. of orthocentre)2+1
P.V. of orthocentre=a+b+c2

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