Question

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.

Answer 1

Let `bara, barb, barc` and `barg` be the position vectors of A, B, C and G respectively.

Then, `bara = 3hati + hatj + 4hatk, barb = - 4hati + 5hatj - 3hatk and barg = - hati + 2hatj + hatk`.

Since G is the centroid of the ΔABC,

By the centroid formula,

`barg = (bara + barb + barc)/3`

∴ `3barg = bara + barb + barc`

∴ `3(- hati + 2hatj + hatk) = (3hati + hatj + 4hatk) + (- 4hati + 5hatj - 3hatk) + barc`

∴ `- 3hati + 6hatj + 3hatk = 3hati + hatj + 4hatk - 4hati + 5hatj - 3hatk + barc`

∴ `- 3hati + 6hatj + 3hatk = (- hati + 6hatj + hatk) + barc`

∴ `barc = - 3hati + 6hatj + 3hatk - (- hati + 6hatj + hatk)`

∴ `barc = - 3hati + 6hatj + 3hatk + hati - 6hatj - hatk`

∴ `barc = - 2hati + 0.hatj + 2hatk`

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