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Question

Show that the following points are coplanar.
(i) (0, −1, 0), (2, 1, −1), (1, 1, 1) and (3, 3, 0)
(ii) (0, 4, 3), (−1, −5, −3), (−2, −2, 1) and (1, 1, −1)

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Solution

(i) The equation of the plane passing through points (0, −1, 0), (2, 1, −1), (1, 1, 1) is given by

x-0y+1z-02-01+1-1-01-01+11-0=0xy+1z22-1121=04x-3 y+1+2 z=04x-3y+2z-3=0Substituting the last point (3, 3, 0) (it means x = 3; y = 3; z = 0) in this plane equation, we get 4 3-3 3+2 0-3 = 012-12 = 00 = 0So, the plane equation is satisfied by the point (3, 3, 0).So, the given points are coplanar.

(ii) The equation of the plane passing through (0, 4, 3), (−1, −5, −3), (−2, −2, 1) is
x-0y-4z-3-1-0-5-4-3-3-2-0-2-41-3=0xy-4z-3-1-9-6-2-6-2=0-18x+10 y-4-12 z-3=09x-5 y-4+6 z-3=09x-5y+6z+2=0Substituting the last point (1, 1, -1) (it means x = 1; y = 1; z=-1) in this plane equation, we get 9 1-5 1+6 -1+2=04-4=00=0So, the plane equation is satisfied by the point (1, 1, -1).So, the given points are coplanar.

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