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

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!