Question

If A(5, 1, p), B(1, q, p) and C(1, −2, 3) are vertices of triangle and `"G"("r", -4/3, 1/3)` is its centroid then find the values of p, q and r

Answer 1

Let `bar"a", bar"b", bar"c"` be the position vectors of points A, B, C respectively of ∆ABC and `bar"g"` be the position vector of its centroid G.

∴ `bar"a" = 5hat"i" + hat"j" + "p"hat"k"`,

`bar"b" = hat"i" + "q"hat"j" + "p"hat"k"`,

`bar"c" = hat"i" - 2hat"j" + 3hat"k"`

and 

`bar"g" = "r"hat"i" - 4/3hat"j" + 1/3hat"k"`

∴ By using centroid formula,

`bar"g" = (bar"a" + bar"b" + bar"c")/3`

∴ `3bar"g" = bar"a" + bar"b" + bar"c"`

∴ `3("r"hat"i" - 4/3hat"j" + 1/3hat"k") = (5hat"i" + hat"j" + "p"hat"k") + (hat"i" + "q"hat"j" + "p"hat"k") + (hat"i" - 2hat"j + 3hat"k")`

∴ `3"r"hat"i" - 4hat"j" + hat"k" = 7hat"i" + ("q" - 1)hat"j" + (2"p" + 3)hat"k"`

∴ By equality of vectors, we get

3r = 7, – 4 = q – 1 and 1 = 2p + 3

∴ r = `7/3`, q = – 3 and p = – 1