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09.16 - [Collect homework#2[Collect summaries of...

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[Collect homework #2] [Collect summaries of section] [Hand out questions for diagnostic quiz #2] Why is it that if a ( b × c ) = 0, then a , b , and c are coplanar? ..?.. b × c must be perpendicular to a , and since b × c is also perpendicular to b and c , the vectors a , b , and c are all perpendicular to b × c ; so a , b , and c must be coplanar (specifically, they lie in the plane perpendicular to b × c ). This analysis presupposes that b × c is not the zero vector. How do we prove the claim in the case where b × c is the zero vector? ..?.. b and c and parallel (they lie on a line), so a , b , and c lie in a plane. Conversely, if a , b , and c are coplanar, a ( b × c ) must be 0, as we’ll now see. The triple product: a ( b × c ) = the signed volume V of the parallelepiped spanned by a , b , and c (which vanishes if a , b , and c are
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coplanar). More precisely, the triple product equals + V or – V , according to whether … ..?.. a , b , and c form a right-handed triple or a left-handed triple.
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