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 KLMN.

Let `bar"k", bar"l", bar"m",bar"n"` be the position vectors of the points K, L, M, N respectively w.r.t. the origin O.

Then, `bar"k" = 5hat"i" - 7hat"j" + 0hat"k"`

`bar"l" = hat"i" + 5hat"j" + 3hat"k"`

`bar"m" = 4hat"i" - 6hat"j" + 3hat"k"`

`bar"n" = 6hat"i" - 4hat"j" + 2hat"k"`

Let G`(bar"g")` be the centroid of the tetrahedron.

Then by centroid formula

`bar"g" = (bar"k" + bar"l" + bar"m" + bar"n")/4`

`= 1/4 [(5hat"i" - 7hat"j" + 0.hat"k") + (hat"i" + 5hat"j" + 3hat"k") + (4hat"i" - 6hat"j" + 3hat"k") + (6hat"i" - 4hat"j" + 2hat"k")]`

`= (16hat"i" - 12hat"j" + 8hat"k")/4`

`= 4hat"i" - 3hat"j" + 2hat"k"`

Hence, the centroid of the tetrahedron is

G ≡ (4, −3, 2)