Solución

Supongamos que p\bold{p} representa la proyección de v\bold{v} sobre u\bold{u}:

p=proyuv=u⋅v∥u∥2u=⟨2,3,2⟩⋅⟨8,−3,−3⟩∥⟨2,3,2⟩∥2⟨2,3,2⟩=16−9−622+32+22⟨2,3,2⟩=117⟨2,3,2⟩=⟨217,317,217⟩\begin{aligned} \bold{p} &= proy_\bold{u}\bold{v}\\ &= \frac{\bold{u}\cdot\bold{v}}{\|\bold{u}\|^2}\bold{u}\\ &= \frac{\lang 2,3,2\rang\cdot\lang 8,−3,−3\rang}{\|\lang 2,3,2\rang\|^2}\lang 2,3,2\rang\\ &= \frac{16-9-6}{2^2+3^2+2^2}\lang 2,3,2\rang\\ &= \frac{1}{17}\lang 2,3,2\rang\\ &=\bigg\lang \frac{2}{17}, \frac{3}{17}, \frac{2}{17}\bigg\rang \end{aligned}

luego

q=v−p=⟨8,−3,−3⟩−⟨217,317,217⟩=⟨13417,−5417,−5317⟩\bold{q} = \bold{v} - \bold{p} = \lang 8,-3,-3\rang - \bigg\lang \frac{2}{17}, \frac{3}{17}, \frac{2}{17}\bigg\rang = \bigg\lang \frac{134}{17}, -\frac{54}{17}, -\frac{53}{17}\bigg\rang

Para verificar nuestro trabajo, podemos usar el producto punto para verificar que p\bold{p} y q\bold{q} son vectores ortogonales:

p⋅q=⟨217,317,217⟩⋅⟨13417,−5417,−5317⟩=26817−16217−10617=0\bold{p}\cdot\bold{q} = \bigg\lang \frac{2}{17}, \frac{3}{17}, \frac{2}{17}\bigg\rang\cdot\bigg\lang \frac{134}{17}, -\frac{54}{17}, -\frac{53}{17}\bigg\rang = \frac{268}{17} - \frac{162}{17} - \frac{106}{17} = 0

entonces

v=p+q=⟨217,317,217⟩+⟨13417,−5417,−5317⟩\bold{v} = \bold{p} + \bold{q} = \bigg\lang \frac{2}{17}, \frac{3}{17}, \frac{2}{17}\bigg\rang + \bigg\lang \frac{134}{17}, -\frac{54}{17}, -\frac{53}{17}\bigg\rang