Given a vector space V, two vectos subspaces U and W are called complementary when their direct sum is V.

$V=\color{red}U\color{black}\oplus \color{blue}W\color{black}$

$\vec v=\color{red}\vec u\color{black}+\color{blue}\vec w\color{black}$

The vector $\color{red}\vec u\color{black}$ is called proyecton of $\vec{v}$ onto $\color{red}U\color{black}$ along $\color{blue}W\color{black}$.

El vector $\color{blue}\vec{w}\color{black}$ is called proyection de $\vec{v}$ onto $\color{blue}W\color{black}$ along $\color{red}U\color{black}$.

 The drawing exemplifies this fact. $\color{red}U\color{black}$ and $\color{blue}W\color{black}$ are complementary subspaces corresponding, respectively, to a plane and a line of $\mathbb{R}^3$. You can modify the vector $\vec v$ and see how its projections vary.

Note that the vector $\color{red}\vec u\color{black}$ is constrcuted by interesecting the plane $\color{red}U\color{black}$  with the line parallel to $\color{blue}W\color{black}$ passing through $\vec v$. Conversely, the vector $\color{blue}\vec{w}\color{black}$ is constructed by intersecting with the line $\color{blue}W\color{black}$ the plane parallel to $\color{red} U\color{black}$ passing through $\vec{v}$.

February 2017. Dynamic sheet developed by Luis Fuentes García  with Geogebra.