If G(a, 2, −1) is the centroid of the triangle with vertices P(1, 2, 3), Q(3, b, −4) and R(5, 1, c) then find the values of a, b and c

#### Solution

Let `bar"p", bar"q", bar"r"` be the position vectors of points P, Q, R respectively of ∆PQR and g be the position vector of its centroid G.

∴ `bar"p" = hat"i" + 2hat"j" + 3hat"k", bar"q" = 3hat"i" + "b"hat"j" - 4hat"k", bar"r" = 5hat"i" + hat"j" + "c"hat"k"` and `bar"g" = "a"hat"i" + 2hat"j" - hat"k"`

∴ By using centroid formula,

`bar"g" = (bar"p" + bar"q" + bar"r")/3`

∴ `3bar"g" = bar"p" + bar"q" + "r"`

∴ `3("a"hat"i" + 2hat"j" - hat"k") = (hat"i" + 2hat"j" + 3hat"k") + (3hat"i" + "b"hat"j" - 4hat"k") + (5hat"i" + hat"j" + "c"hat"k")`

∴ `3"a"hat"i" + 6hat"j" - 3hat"k" = (1 + 3 + 5)hat"i" + (2 + "b" + 1)hat"j" + (3 - 4 + "c")hat"k"`

∴ `3"a"hat"i" + 6hat"j" - 3hat"k" = 9hat"i" + ("b" + 3)hat"j" + ("c" - 1)hat"k"`

∴ By equality of vectors, we get

3a = 9, 6 = b + 3 and −3 = c − 1

∴ a = 3, b = 3 and c = −2