Cross product

For cross product in 2-D, it represents the area of parallelogram. And it has an order that i on the right of the j.

Actually, cross product is in 3-D, and the result of the cross product is a vector with length of the area of parallelogram, perpendicular to the parallelogram obeying right hand rule.

Cross product can be calculated as follow.

Geometry

By using duality, exist a vector p dot product [x, y, z] equals determinant.

3-D determinant means the volumn of the cube, we need to calculate component of [x, y, z] perpendicular to v and w times area of parallelogram.

This equals to the vector p with the length of area of parallelogram by using duality.

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Reference
https://www.3blue1brown.com/topics/linear-algebra