Question

Calcule um vetor que seja ortogonal a u=(1,0,1) e v=(1,-1,0) e que tenha norma igual a 5

Answer 1

Peguemos o vetor (x,y,z) sabemos que:

$x+z=0$
$x-y=0$
$\sqrt{x^2+y^2+z^2}=5$

$x=-z$
$y=-z$
$\sqrt{3z^2}=5$
$3z^2=25$
$z^2=\frac{25}{3}$
$z=\pm\frac{5}{\sqrt3}=\pm\frac{5\sqrt3}{3}$

Entao os vetores podem ser:

$\left[\begin{array}{c}-\frac{5\sqrt3}{3}&-\frac{5\sqrt3}{3}&\frac{5\sqrt3}{3}\end{array}\right] \ ou\ \left[\begin{array}{c}\frac{5\sqrt3}{3}&\frac{5\sqrt3}{3}&-\frac{5\sqrt3}{3}\end{array}\right]$