A, B C and D are four points in a plane with position vectors, `veca, vecb vecc and vecd` respectively, such that `(veca-vecd).(vecb-vecc)= (vecb-vecd).(vecc-veca)=0` then point D is the ______ of triangle ABC.

Answer

orthocenter

Explanation

Given that `veca, vecb , vecc and vecd` are position vectors of points A,B,C and D, respectively, such that
`(veca - vecd) . (vecb - vecc) = (vecb.vecd) . (vecc- veca) =0`
`Rightarrow vec(DA).vec(CB) = vec(DB).vec(AC) =0`
` Rightarrow vec(DA) bot vec(CB) and vec(DB) bot vec(AC)`
Clerly, D is the orthocentre of `triangleABC`