Question

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

Answer 1

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!