If the origin is the centroid of the triangle whose vertices are A (2,p,-3), B(q,-2,5) and C(-5,1,r), then find the values of p,q and r.

Explanation

Let `bar a, barb, barc` be the position vectors of the vertices A, B and C respectively.
`:. " " bara = 2 hati+phatj-3hatk,`
` barb = qhati - 2 hatj + 5 hat k,`
`barc = - 5 hati +hatj + rhatk ` ...(1)
G `(0, 0, 0) ` is the centroid of the triangle, then position vector
` barg = 0 hati + 0 hatj +0 hat k`. ....(2)
` :. ` By centroid formula , ` barg = (bara + barb+bar c)/3 .`
` rArr " " 0hati + 0 hatj + 0 hatk`
` = ((2hati + p hatj-3hatk)+(qhati- 2hatj+5hatk)+(-5hati+hatj+rhatk))/3 `
{from eqn. (1) and (2)]
`rArr 0 hati + 0 hatj + 0 hatk`
` = ((2 + q - 5)hati+(p-2+1) hatj+(-3+5+r)hatk)/3`
` rArr 0hati+0 hatj+0hatk = (-3+q)hati+(p-1)hatj + (2 + r)hatk`
`:. " " -3 + q = 0, p - 1 = 0, 2 + r = 0`
` :. " " q = 3,p=1, r = - 2`
Hence , p =1 , q = 3, r = - 2