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Under what condition three vectors A, B, C will be co-planer? I want to learn this rules and theorem.

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Since it is a homework question i will just give hints. Hint no 1: If you take cross product between two vectors you get a resultant vector which is perpendicular to the plane where the two vectors are.

Hint no 2: Dot product between two mutually perpendicular vectors is zero.

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A totally antisymmetric product of three vectors will give zero if they are zero, colinear, or coplanar and if any of those three things happen then they are coplanar. Basically an antisymmetric product of three vectors gives the oriented 3-volume they span, and if coplanar, zero is all you can get.

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  • $\begingroup$ This answer is correct, but level of sophistication is, I think, beyond the spirit of the original post. $\endgroup$
    – garyp
    Jan 19 '15 at 19:24
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In $\mathbb{R}^3$, three vectors are co-planar if they are linearly dependent, i.e. if there is a solution to the equation $C=s \cdot A + t \cdot B$.

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