Show that if ~w is orthogonal to ~u and ~v, then ~w is orthogonal to every vector ~x in Span{~u, ~v}.

Explanation:[tex] \overline{w} [/tex] is ortogonal to a vector [tex] \overline{c} [/tex] if, and only if, the scalar product [tex] < \overline{w},\overline{c} > = 0. [/tex] Hence, it should be [tex]< \overline{w},\overline{u} > = < \overline{w},\overline{v} > = 0 . [/tex]The scalar product is linear, so it takes constants and sums out. If [tex] \overline{x} [/tex] is a vector spanned by [tex] \overline{u} [/tex] and [tex] \overline{w} ,  [/tex] lets say [tex] \overline{x} = a*\overline{u} + b*\overline{v} , [/tex] for certain complex (or real) values a and b, then we have[tex] < \overline{w},\overline{x} > = < \overline{w}, a*\overline{u} + b*\overline{v}> = a * < \overline{w},\overline{u} > + b * < \overline{w},\overline{v} > = a*0+b*0 = 0 [/tex]Because both [tex]< \overline{w},\overline{u} > [/tex] and [tex]< \overline{w},\overline{v} > [/tex] are equal to 0. That proves that [tex] \overline{x} [/tex] , an arbitrary element in [tex] Span\{\overline{u}, \overline{v}\} , [/tex] is perpendicular to [tex] \overline{w} [/tex] .I hope that helped you!