If two of the vertices of a triangle are A (3, 1, 4) and B(− 4, 5, −3) and the centroid of the triangle is at G (−1, 2, 1), then find the coordinates of the third vertex C of the triangle

#### Solution

Let `bar"a", bar"b", bar"c"` and `bar"g"` be the position vectors of A, B, C and G respectively.

Then, `bar"a" = 3hat"i" + hat"j" + 4hat"k",

bar"b" = - 4hat"i" + 5hat"j" - 3hat"k"` and

`bar"g" = - hat"i" + 2hat"j" + hat"k"`.

Since G is the centroid of the ΔABC,

By the centroid formula,

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

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

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

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

= `-3hat"i" + 6hat"j" + 3hat"k" - 3hat"i" - hat"j" - 4hat"k" + 4hat"i" - 5hat"j" + 3hat"k"`

∴ `bar"c" = - 2hat"i" + 0.hat"j" + 2hat"k"`

∴ The coordinates of third vertex C are (−2, 0, 2).