Question

If the centroid of a tetrahedron OABC is (1, 2, - 1) where A(a, 2, 3), B(1, b, 2), C(2, 1, c), find the distance of P(a, b, c) from origin.

Answer 1

Let G = (1, 2, -1) be the centroid of the tetrahedron OABC.

Let `bar"a", bar"b", bar"c", bar"g"` be the position vectors of the points A, B, C, G respectively w.r.t. O.

then `bar"a" = "a"hat"i" + 2hat"j" + 3hat"k"`,
`bar"b" = hat"i" + "b"hat"j" + 2hat"k", `
`bar"c" = 2hat"i" + hat"j" + "c"hat"k",` 
`bar"g" = hat"i" + 2hat"j" - hat"k"`

By formula of centroid of a tetrahedron,

`bar"g" = (bar"0" + bar"a" + bar"b" + bar"c")/4`

∴ `4bar"g" = bar"a" + bar"b" + bar"c"`

∴ `4(hat"i" + 2hat"j" - hat"k") = ("a"hat"i" + 2hat"j" + 3hat"k") + (hat"i" + "b"hat"j" + 2hat"k") + (2hat"i" + hat"j" + "c"hat"k")`

`∴ 4hat"i" + 8hat"j" - 4hat"k" = ("a" + 1 + 2)hat"i" + (2 + "b" + 1)hat"j" + (3 + 2 + "c")hat"k"`

∴ `4hat"i" + 8hat"j" - 4hat"k" = ("a" + 3)hat"i" + ("b" + 3)hat"j" + ("c" + 5)hat"k"`

By equality of vectors

a + 3 = 4, b + 3 = 8, c + 5 = - 4

∴ a = 1, b = 5, c = - 9

∴ P = (a, b, c) = (1, 5, -9)

Distance of P from origin = `sqrt(1^2 + 5^2 + (- 9)^2)`

`= sqrt(1 + 25 + 81)`

`= sqrt107` units