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Question
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, r.
Solution
Let `bara, barb,barc` be the position vectors of `triangle ABC` whose vertices are A(2, p, –3), B(q, –2, 5) and C(–5, 1, r)
`:.bara = 2hati + pbarj - 3bark`, `barb = qbari - 2barj + 5bark`, `barc = -5bari + barj + rbark`
Given that origin O is the centroid of `triangle ABC`
`:. barO = (bara+barb+barc)/3`
`:. bara+barb+barc = barO`
`2hati + phatj - 3hatk + hatj - 2hatj +5hatk - 5hati + hatj + rhatk = barO`
`=> (2+q-5)hati + (p-2+1)hatj + (-3+5+r)hatk = 0hati + 0hatj + 0hatk`
by equality of vectors
2 + q - 5 = 0 ⇒ q = 3
p - 2 + 1 = 0 ⇒ p = 1
-3 + 5 + r = 0 ⇒ r = -2
∴ p = 1, q = 3 and r = -2
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