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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.
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"`
∴ `3(- hat"i" + 2hat"j" + hat"k") = (3hat"i" + hat"j" + 4hat"k") + (- 4hat"i" + 5hat"j" - 3hat"k") + bar"c"`
∴ `- 3hat"i" + 6hat"j" + 3hat"k" = 3hat"i" + hat"j" + 4hat"k" - 4hat"i" + 5hat"j" - 3hat"k" + bar"c"`
∴ `- 3hat"i" + 6hat"j" + 3hat"k" = (- hat"i" + 6hat"j" + hat"k") + bar"c"`
∴ `bar"c" = - 3hat"i" + 6hat"j" + 3hat"k" - (- hat"i" + 6hat"j" + hat"k")`
∴ `bar"c" = - 3hat"i" + 6hat"j" + 3hat"k" + hat"i" - 6hat"j" - hat"k"`
∴ `bar"c" = - 2hat"i" + 0.hat"j" + 2hat"k"`
∴ The coordinates of third vertex C are (−2, 0, 2).
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