Determine whether the points ଵ (−6, 4, 8), ଶ(9, −2, 0) and ଷ

1.	Determine whether the points ଵ (−6, 4, 8), ଶ(9, −2, 0) and ଷ(1, −5, 3) lie on the same line.
Answer» A = (-6,4,8), B = (9,-2,0), C = (1, -5,3) AB = \(\sqrt{(9-(-6))^2+(-2-4)^2+(0-8)^2}\) = \(\sqrt{15^2+(-6)^2+(-8)^2}\) = \(\sqrt{225+36+64}\) = \(\sqrt{325}\) = \(5\sqrt13\) = 18.028 BC = \(\sqrt{(1-9)^2+(-5+2)^2+(3-0)^2}\) = \(\sqrt{8^2+3^2+3^2}\) = \(\sqrt{64+9+9}\) = \(\sqrt{82}\) = 9.055 AC = \(\sqrt{1-(-6))^2+(-5-4)^2+(3-8)^2}\) = \(\sqrt{49+81+25}\) = \(\sqrt{155}\) = 12.449 Clearly, AB \(\ne\)BC + AC Or AC \(\ne\) AB + BC Or BC \(\ne\) AB + AC Therefore, points A, B and C are not collinear(not on same plane) ,they forms a triangle.

Determine whether the points ଵ (−6, 4, 8), ଶ(9, −2, 0) and ଷ(1, −5, 3) lie on the same line.

Answer»

A = (-6,4,8), B = (9,-2,0), C = (1, -5,3)

AB = \(\sqrt{(9-(-6))^2+(-2-4)^2+(0-8)^2}\)

= \(\sqrt{15^2+(-6)^2+(-8)^2}\)

= \(\sqrt{225+36+64}\)

= \(\sqrt{325}\) = \(5\sqrt13\) = 18.028

BC = \(\sqrt{(1-9)^2+(-5+2)^2+(3-0)^2}\)

= \(\sqrt{8^2+3^2+3^2}\)

= \(\sqrt{64+9+9}\) = \(\sqrt{82}\) = 9.055

AC = \(\sqrt{1-(-6))^2+(-5-4)^2+(3-8)^2}\)

= \(\sqrt{49+81+25}\)

= \(\sqrt{155}\) = 12.449

Clearly, AB \(\ne\)BC + AC

AC \(\ne\) AB + BC

BC \(\ne\) AB + AC

Therefore, points A, B and C are not collinear(not on same plane) ,they forms a triangle.

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