If a,b,c are three non coplanar vectors, then prove that the vector equation r = ( 1 − p − q ) a → + p b → + q c → represents a planeLet r = x a → + y b → + z c → Comparing with the given equation, we obtainx+y+z=1which is a planeWhat I don’t understand is how does this say r is a plane, since r is actually x a → + y b → + z c →

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If a,b,c are three non coplanar vectors, then prove that the vector equation   r  =  (  1  −  p  −  q  )            a      →        +  p            b      →        +  q            c      →       represents a planeLet   r  =  x            a      →        +  y            b      →        +  z            c      →      Comparing with the given equation, we obtainx+y+z=1which is a planeWhat I don’t understand is how does this say r is a plane, since r is actually   x            a      →        +  y            b      →        +  z            c      →

honotMornne · Accepted Answer

Rearrange to            r      →        =            a      →        +  p  (            b      →        −            a      →        )  +  q  (            c      →        −            a      →        )And this is of the form of the plane equation            r      →        =            p      →        +  λ            s      →        +  μ            q      →      More precisely, this is the plane that passes through the point with position vector             a      →        and with normal vector   (            b      →        −            a      →        )  ×  (            c      →        −            a      →        )

robbbiehu · Accepted Answer

A plane             r      →        =  (  x  ,  y  ,  z  ) that passes the point             a      →       and with normal vector             n      →       is                    (1)                    (                              r            →                          −                              a            →                          )        ⋅                              n            →                          =        0            Observe that the given vector can be rearranged as            r      →        −            a      →        =  p  (            b      →        −            a      →        )  +  q  (            c      →        −            a      →        )which satisfies the plane equation (1) with the choice of             n      →        =  (            b      →        −            a      →        )  ×  (            c      →        −            a      →        )

If a,b,c are three non coplanar vectors, then prove that the vector equation r=(1−p−q)vec(a)+p vec(b)+q vec(c) represents a plane

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