## Normal vector as a perpendicular vector of the surface tangent vectors

Normal vector has the same direction to the cross product of two tangent vectors of a surface. Figure 2 shows the tangent vectors are correctly transformed by the matrix

*M*that magnifies only

*x*direction. However, their cross product is not necessary to the same as the transformation of normal by the matrix

*M*.

Figure 2. The normal vector is a cross product of tangent vectors n and u. Tangent vectors are linear to vM, but not for the normal vector. |

**,**

*u***can be transformed by**

*v**M*, but their cross product is not. In general,

Are you convinced this is the reason distinguishing a usual vector and a normal vector? If you think about the

*x*component of the cross product,

*uy vz - uz vy*, this is not linear. Therefore, a linear transformation cannot transform this. This is my first explanation.

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