[NOTES/EM-07021] Cross Product Rule

Convention for normal to the plane of paper
When drawn from a point two vectors define a plane. The cross product will be perpendicular to the plane containing the the two vectors. We state a convention for  direction of vectors perpendicular to the plane of paper. There are two possibilities. A normal vector can be out of the plane, or into the plane of the paper. A circle with a dot. \(\odot\), or dots in a region will be used for the direction out of the plane. A cross \(\otimes\), or into sign \(\times\) will be used to indicate the direction into the plane of the paper. \\



Cross product rule
A popular way of giving the direction is given in the screen shot, \Figref{em-shot-07001}, of a \href{https://youtu.be/gPnWm-IXoAY?t=159}{youtube video}



Explanation of right hand rule in \Figref{em-shot-07001} is as follows. Let your fingers point in the direction of the first vector, \(\vec{A}\) and curl around via the smaller angle towards the second vector \(\vec{B}\). Then direction of thumb indicates the direction of the vector product \(\vec{A}\times \vec{B}\). You can find some more popular rules to determine the direction of the cross product. In the figures Fig.1 and Fig.2 are shown  two possibilities relative directions   of vectors \(\vec A, \vec B\) and corresponding cross product \(\vec A\times \vec B\), drawn as arrows in red, are sketched. The rules for determining direction of cross product of two vectors are as follows.

If an anticlockwise rotation aligns A ~ with vector B, ~ as in Fig.1(a), the cross product is out of the plane of the paper.	If a clockwise rotation aligns A ~ with vector B, ~ as in Fig.1(b), the cross product is into the plane of the paper.

Thus a way to remember the cross product rule is coded in the following diagram.

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