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

User Ossama
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Step-by-step explanation:


\overline{w} is ortogonal to a vector
\overline{c} if, and only if, the scalar product
< \overline{w},\overline{c} > = 0. Hence, it should be
< \overline{w},\overline{u} > = < \overline{w},\overline{v} > = 0 .

The scalar product is linear, so it takes constants and sums out. If
\overline{x} is a vector spanned by
\overline{u} and
\overline{w} , lets say
\overline{x} = a*\overline{u} + b*\overline{v} , for certain complex (or real) values a and b, then we have


< \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

Because both
< \overline{w},\overline{u} > and
< \overline{w},\overline{v} > are equal to 0. That proves that
\overline{x} , an arbitrary element in
Span\{\overline{u}, \overline{v}\} , is perpendicular to
\overline{w} .

I hope that helped you!

User Authchir
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