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Question
Show that the points A (1, 2, 7), B (2, 6, 3) and C (3, 10, –1) are collinear.
Solution
The given points are A(1, 2, 7), B(2, 6, 3) and C(3, 10, -1).
∴ `vec(AB) = (2 - 1)hati + (6 - 2)hatj + (3 - 7)hatk = hati + 4hatj - 4hatk`
`vec(BC) = (3 - 2)hati + (10 - 6)hatj + (-1 - 3)hatk = hati + 4hatj - 4hatk`
`vec(AC) = (3 - 1)hati + (10 - 2)hatj + (-1 - 7)hatk = 2hati + 8hatj - 8hatk`
`|vec(AB)| = sqrt(1^2 + 4^2 + (-4)^2) `
`= sqrt(1 + 16 + 16) `
`= sqrt33`
`|vec(BC)| = sqrt(1^2 + 4^2 + (-4)^2)`
` = sqrt(1 + 16 + 16) `
`= sqrt33`
`|vec(AC)| = sqrt(2^2 + 8^2 + 8^2)`
` = sqrt(4 + 64 + 64) = sqrt(132) `
`= 2sqrt33`
∴ `|vec(AC)| = |vec(AB)| + vec(BC)`
Hence, the given points A, B and C are collinear.
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