How have the right to he prove the clues \$A=(a_1, a_2, a_3)\$, \$B=(b_1, b_2, b_3)\$, \$C=(c_1, c_2, c_3)\$, \$D=(d_1, d_2, d_3)\$ belong to the same plane, as if they belong deserve to then find plane. I know how to prove three offered points belong the exact same plane.

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Take any type of three that the points and determine the equation of the plane. As TonyK said, 3 points always belong to one airplane and, if they execute not every lie in a line, then the identify a unique plane. Once you have the equation the the plane, placed the collaborates of the fourth suggest into the equation to check out if it is satisfied. If the 3 points you made decision do occur to lie on a single line then you room done- any type of fourth point will identify a airplane that all four points lie on.

Hint: analyze by \$-A\$ so the we have the right to assume \$A=(0,0,0)\$. Climate points \$A,B,C,D\$ lied on a aircraft iff vectors \$B,C,D\$ execute not span whole space, i.e. If \$\$left| eginarraycccb_1 & b_2 & b_3 \c_1 & c_2 & c_3 \d_1 & d_2 & d_3 endarray ight|=0\$\$

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