\title{Proyección ortogonal 2}
\language{fr}
\range{-5..5}
\author{Adeline GRELOT et Bernadette PERRIN-RIOU}
\email{bpr@math.u-psud.fr}
\computeanswer{yes}
\format{html}
\precision{10000}

\integer{n=3}

\integer{a=random(randint(-5..-1),randint(1..5))}
\integer{b=random(randint(-5..-1),randint(1..5))}
\integer{c=random(randint(-5..-1),randint(1..5))}

\text{v3=random(randint(1..2), randint(-2..-1)),random(randint(1..2), randint(-2..-1)),random(randint(1..2), randint(-2..-1))}

Calcul de la normale
\text{normale=\a,\b,\c}

Projection de v3 sur la normale

\rational{prod=simplify((item(1,\v3))*(item(1,\normale))+(item(2,\v3))*(item(2,\normale))+(item(3,\v3))*(item(3,\normale)))}
\rational{norm2=simplify((item(1,\normale))^2+(item(2,\normale))^2+(item(3,\normale))^2)}
\rational{sol1=item(1,\v3)-(\prod)/\norm2*(item(1,\normale))}
\rational{sol2=item(2,\v3)-(\prod)/\norm2*(item(2,\normale))}
\rational{sol3=item(3,\v3)-(\prod)/\norm2*(item(3,\normale))}

\text{sol=\sol1,\sol2,\sol3}
\text{equation=simplify((\a)*x + (\b)*y + (\c)*z)}

\statement{
Sea \(\RR^3) con el producto escalar usual y \(F) el subespacio vectorial de ecuación <p><center>\(\equation=0). </center></p>
Hallar la proyección ortogonal del vector (\v3) sobre \(F).}

\answer{proyección ortogonal}{\sol}{type=vector}