Suppose that y is orthogonal to u v and w Show that y lies i

Suppose that y is orthogonal to u, v, and w. Show that y lies in the orthogonal complement of span{u, v, w}.

Solution

An arbitrary vector x in the span{u,v,w} being a linear combination of u,v,w is of the form au+bv+cw where a,b,c are abitrary scalars. Then y. x = y. (au+bv+cw) = y.au +y.bv +y.cw = a(y.u)+b(y.v)+c(y.w) = a.0+ b.0+c.0 = 0. Therefore, y is orthogonal to x and therefore, y lies in the orthogonal complement of span{u,v,w}.