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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
Solution
Let `veca, vecb, vecc` be the position vectors of points A, B, C respectively of ∆ABC and `vecG` be the position vector of its centroid G.
∴ `veca = 5hati + hatj + phatk`,
`vecb = hati + qhatj + phatk`,
`vecc = hati - 2hatj + 3hatk`
And `vecG = rhati - 4/3hatj + 1/3hatk`
∴ By using centroid formula,
`vecG = (veca + vecb + vecc)/3`
∴ `3vecG = veca + vecb + vecc`
∴ `3(rhati - 4/3hatj + 1/3hatk) = (5hati + hatj + phatk) + (hati + qhatj + phatk) + (hati - 2hatj + 3hatk)`
∴ `3rhati - 4hatj + hatk = 7hati + (q - 1)hatj + (2p + 3)hatk`
∴ By equality of vectors, we get
3r = 7, – 4 = q – 1 and 1 = 2p + 3
⇒ r = `7/3`, q = – 3 and p = – 1
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