Question

Find the centroid of tetrahedron with vertices K(5, −7, 0), L(1, 5, 3), M(4, −6, 3), N(6, −4, 2)

Answer 1

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 = (veck + 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).