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Question
Find the centroid of tetrahedron with vertices K(5, −7, 0), L(1, 5, 3), M(4, −6, 3), N(6, −4, 2)
Solution
Let G be the centroid of the tetrahedron K, L, M, N.
Let `vecp, vecl, vecm, vecn` be the position vectors of the points K, L, M, N respectively w.r.t. the origin O.
Then, `vecp = 5hati - 7hatj + 0hatk`
`vecl = hati + 5hatj + 3hatk`
`vecm = 4hati - 6hatj + 3hatk`
`vecn = 6hati - 4hatj + 2hatk`
Let G(g) be the centroid of the tetrahedron.
Then by centroid formula
`vecg = (vecp + vecl + vecm + vecn)/4`
= `1/4 [(5hati - 7hatj + 0.hatk) + (hati + 5hatj + 3hatk) + (4hati - 6hatj + 3hatk) + (6hati - 4hatj + 2hatk)]`
= `1/4(16hat"i" - 12hat"j" + 8hat"k")`
= `4hati - 3hatj + 2hatk`
Hence, the centroid of the tetrahedron is G = (4, −3, 2).
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