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in 3D Coordinate Geometry by (3.3k points)
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Using distance formula show that points A(1, −1, 3), B(2, −4, 5) and C(5, −13, 11) are collinear.

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Best answer

AB =

\((\sqrt{{(1-2)^{2}}+(-1+4)^{2}+(3-5)^{2}}\\=\sqrt{1+9+4}\\=\sqrt{14}\)

BC =

\((\sqrt{{(2-5)^{2}}+(-4+13)^{2}+(5-11)^{2}}\\=\sqrt{9+81+36}\\=3\sqrt{14}\)

CA =

\((\sqrt{{(5-1)^{2}}+(-13+1)^{2}+(11-3)^{2}}\\=\sqrt{16+144+64}\\=\sqrt{224}=4\sqrt{14}\)

Since AB + BC = CA ∴ A, B, C are collinear

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