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  4. If G( vecg), H( vech) and P( vecp) are centroid, orthocenter and circumcenter of a triangle and x vecp+y vech+z vecg = 0 then (x, y, z) =

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Q. If $G\left(\vec{g}\right), H\left(\vec{h}\right)$ and $P\left(\vec{p}\right) $ are centroid, orthocenter and circumcenter of a triangle and $x\vec{p}+y\vec{h}+z\vec{g} = 0 $ then $(x, y, z) =$ _____________

MHT CETMHT CET 2016Vector Algebra

Solution:

We know that, orthocentre, centroid and circumcentre of a triangle are collinear and centroid divides orthocentre and circumcentr in the ratio 2: 1
By using internally division,
$\frac{2 \vec{p}+1 \vec{h}}{2+1} =g $
$\Rightarrow 2 \vec{p}+\vec{h}-3 \vec{g}=0$
But it is given $x \vec{p}+y \vec{h}+z \vec{g}=0$
$x=2, y=1 $ and $ z=-3$