Question

Se u = −8i + 4j e v=7i − 6j, encontre a projeção vetorial de u sobre v e de v sobre u.

Answer 1

u=(-8,4,0)
v=(7,-6,0)
|u|=√(-8)²+4²+0²=√80
|v|=√7²+(-6)²+0²=√85
<u,v>=<v,u>=-80

Então:

$proju_{v} = \frac{\ \textless \ u,v\ \textgreater \ }{|v|^2} v \\ \\ proju_{v} == \frac{-56-24+0}{ (\sqrt{85})^2 }(7,-6,0) \\ \\ proju_{v} = \frac{-80}{85} (7,-6,0) \\ \\ proju_{v} = \frac{-16}{17} (7,-6,0) \\ \\ proju_{v} =(- \frac{112}{17}, \frac{96}{17},0)$

E a proj de v sobre u:

$projv_{u} = \frac{\ \textless \ v,u\ \textgreater \ }{|u|^2} u \\ \\ projv_{u} = \frac{-80}{ (\sqrt{80})^2 } (-8,4,0) \\ \\ projv_{u} = \frac{-80}{80} (-8,4,0) \\ \\ projv_{u} =(8,-4,0)$