Question

Dados os vetores ~u = (3, 0, 1) e ~v = (−2, 1, 2), determinar proj~v~u e proj~u~v.

Answer 1

Basta aplicar a fórmula:

$\text{proj}_uv=\frac{\left \langle u,v \right \rangle}{\left \| u \right \|^2}\cdot u$

$\text{proj}_uv=\frac{\left \langle (3,0,1),(-2,1,2) \right \rangle}{3^2+0^2+1^2}\cdot (3,0,1)$

$\text{proj}_uv=\frac{3\cdot(-2)+0\cdot1+1\cdot2}{10}\cdot (3,0,1)$

$\text{proj}_uv=\frac{-4}{10}\cdot (3,0,1)$

$\text{proj}_uv=-\frac{2}{5}\cdot (3,0,1)$

$\text{proj}_uv=(-\frac{6}{5},0,-\frac{2}{5})$

Invertendo $u$ e $v$:

$\text{proj}_vu=\frac{-4}{(-2)^2+1^2+2^2}\cdot (-2,1,2)$

$\text{proj}_vu=\frac{-4}{9}\cdot (-2,1,2)$

$\text{proj}_vu=(\frac{8}{9},-\frac{4}{9},-\frac{8}{9})$