Is the Triple Scalar Product Always Zero?

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SUMMARY

The discussion clarifies that the triple scalar product of three vectors is zero if and only if all three vectors are coplanar. The user expresses confusion regarding whether two coplanar vectors are sufficient for a zero scalar product, but it is established that the condition requires all three vectors to be coplanar. The textbook confirms this principle, emphasizing that the projection of one vector onto the plane formed by the other two results in a zero value, thus leading to a zero triple scalar product.

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  • Knowledge of coplanarity in three-dimensional space
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Hello,

I am confused how vectors that are coplanar will give a triple product of zero? Or is it the case that all 3 vectors must be coplanar for a triple product of zero, or is 2 sufficient? I.e. the vector being dotted with one of the vectors being crossed in the same plane, will this produce a zero scalar product?
 
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Take any two of those coplanar vectors. The cross product is either zero or is normal to both of those vectors -- and every other vector that is coplanar with those first two vectors. What's the dot product of two vectors that are normal to one another?
 
ok, so I see that the textbook specified that if all 3 vectors are coplanar, then its triple scalar product is zero, which makes sense to me because the projection is going to be zero. It's just that the accompanying figure 3.28 doesn't make me think that the vectors are coplanar.
EDIT: here is the figure that I am referring to
 

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