Solución

Apartado a

Sustituye los componentes de v\bold{v} y u\bold{u} en la fórmula 2.6 para la proyección:

proyuv=u⋅v∥u∥(1∥u∥u)=u⋅v∥u∥2u=⟨−1,4,3⟩⋅⟨3,5,1⟩∥⟨−1,4,3⟩∥2⟨−1,4,3⟩=−3+20+3(−1)2+42+32⟨−1,4,3⟩=2026⟨−1,4,3⟩=(−1013,4013,3013)\begin{aligned} proy_u\bold{v} &= \frac{\bold{u}\cdot\bold{v}}{\|\bold{u}\|}\bigg(\frac{1}{\|\bold{u}\|}u\bigg) = \frac{\bold{u}\cdot\bold{v}}{\|\bold{u}\|^2}u\\ &= \frac{\lang −1,4,3\rang\cdot\lang 3,5,1\rang}{\|\lang −1,4,3\rang\|^2}\lang −1,4,3\rang\\ &= \frac{−3+20+3}{(−1)^2+4^2+3^2}\lang −1,4,3\rang\\ &= \frac{20}{26}\lang −1,4,3\rang\\ &= \bigg(-\frac{10}{13}, \frac{40}{13}, \frac{30}{13}\bigg) \end{aligned}

Apartado b

Para encontrar la proyección bidimensional, simplemente adapta la fórmula al caso bidimensional:

proyuv=u⋅v∥u∥(1∥u∥u)=u⋅v∥u∥2u=(i+6j)⋅(3i−2j)∥i+6j∥2(i+6j)=1(3)+6(−2)12+62(i+6j)=−937(i+6j)=−937i−5437i\begin{aligned} proy_u\bold{v} &= \frac{\bold{u}\cdot\bold{v}}{\|\bold{u}\|}\bigg(\frac{1}{\|\bold{u}\|}u\bigg) = \frac{\bold{u}\cdot\bold{v}}{\|\bold{u}\|^2}u\\ &= \frac{(\bold{i}+6\bold{j})\cdot(3\bold{i}−2\bold{j})}{\|\bold{i}+6\bold{j}\|^2}(\bold{i}+6\bold{j})\\ &= \frac{1(3)+6(−2)}{12+62}(\bold{i}+6\bold{j})\\ &=-\frac{9}{37}(\bold{i}+6\bold{j})\\ &= -\frac{9}{37}\bold{i}-\frac{54}{37}\bold{i} \end{aligned}