Question about right-handed coordinate system

Question

The following question is from the book 'Calculus Two' by Flanigan/Kazdan:

Prove the following assertion: suppose that $X, Y$ and $Z$ are linearly independent. Then $X, Y$ and $Z$ is a right-handed coordinate system if and only if $Z$ and $X \times Y$ lie on the same side of the plane spanned by $X$ and $Y$.

Hint: this last part is difficult because it requires a formulation of a definition of "same side", and then an appropriate use of it.

So I have to show that $\langle X \times Y, Z \rangle > 0$ iff $Z$ and $X \times Y$ lie on the same side of the plane specified. Specifically, as the hint predicts, I don't know how to formulate a definition of "same side". If anyone knows how, I'd be very grateful. Thanks in advance.

Answer 1

Geometrically, a typical definition is:

• If $H$ is a plane and $P,Q$ are two points not lying on $H$, then $P$ and $Q$ lie on the same side of $H$ if and only if the line segment $\overline{PQ}$ does not intersect $H$

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Algebraically, if $H$ is the plane through points $X$, $Y$, and the origin, then in my opinion a more convenient characterization is

• $P$ and $Q$ lie on the same side of the plane if and only if $ \operatorname{det}(P, X, Y)$ and $\operatorname{det}(Q, X, Y) $ have the same sign

Alternatively, the determinants can be expressed by triple products; e.g. $\operatorname{det}(P, X, Y) = P \cdot (X \times Y)$.

You can argue this really does capture the notion of "same side" by noting:

• $\operatorname{det}(R, X, Y)$ is continuous in $R$
• $\operatorname{det}(R, X, Y)$ takes both positive and negative values
• $\operatorname{det}(R, X, Y)$ is zero if and only if $R$ lies on $H$

so the sign of this function really does separate the complement of $H$ into two disconnected parts.

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Another common algebraic definition is to let $N$ be a normal vector to $H$, and then define

• $P$ and $Q$ lie on the same side of $H$ if and only if $P \cdot N$ and $Q \cdot N$ have the same sign.