Utilidades

1. Produto escalar: [; \underline{u} \cdot \underline{v}=\displaystyle \sum_{i=1}^N u_i~v_i ;]
2. Square Length: [; |\underline{u}|^2=\underline{u} \cdot \underline{u}=\displaystyle \sum_{i=1}^N u_i^2 ;]
3. Normalização: [; \underline{e}=\frac{\underline{v}}{ \underline{v}} ;]
4. Projeção de [; \underline{u} ;] em [; \underline{v};]: [; \underline{u}_{\parallel}= \frac { \underline{u} \cdot \underline{v} }{ |\underline{v}|^2 } \underline{v} = (\underline{u} \cdot \underline{e}) \underline{e} ;]
5. Complemento Ortogonal: [; \underline{u}_{\perp}+\underline{u}_{\parallel} = \underline{u} ;] [; \underline{u}_{\perp}= \underline{u}-\underline{u}_{\parallel} = \underline{u}-(\underline{u} \cdot \underline{e}) \underline{e} ;] [; \underline{u}_{\parallel} \cdot \underline{u}_{\perp} = \underline{e} \cdot \underline{u}_{\perp} = \underline{v} \cdot \underline{u}_{\perp} =0 ;]

    def DotProduct(v,w):
        return v*w
    
    def SqLength(v):
        return v*v
    
    def Length(v):
        return sqrt( v.SqLength() )
    
    def Normalize(v,length=1.0):
        w=Vector()
        if (v.Length()>0.0):
            w=v*(1.0/v.Length())
    
    def Hat2(v):
        return Vector([ -v[1],v[0] ])

    def Projection(v,e):
        return e*(v*e)/e.SqLength()
    
    def Complement(v,e):
        return v-v.Projection(e)