Mechanics of Continua and Structures
The component of an arbitrary vector $\boldsymbol{u}$ in the direction of, i.e., parallel to $\boldsymbol{a}$ is
\[\begin{align} \boldsymbol{u}_{\parallel}&=(\boldsymbol{a}\cdot \boldsymbol{u}) \boldsymbol{a}\\ &=(\boldsymbol{a} \otimes \boldsymbol{a})\boldsymbol{u} \end{align}\]The component of an arbitrary vector $\boldsymbol{u}$ that is perpendicular to $\boldsymbol{a}$ is
\[\begin{align} \boldsymbol{u}_{\perp}&=\boldsymbol{u}-\boldsymbol{u}_{\parallel}\\ &=\boldsymbol{u}-(\boldsymbol{a} \otimes \boldsymbol{a})\boldsymbol{u}\\ &=\left(\boldsymbol{I}-\boldsymbol{a} \otimes \boldsymbol{a}\right)\boldsymbol{u} \end{align}\]From the definition of $\boldsymbol{u}_{\perp}$ and $\boldsymbol{u}_{\parallel}$ it follows that \(\boldsymbol{u}=\boldsymbol{u}_{\perp}+\boldsymbol{u}_{\parallel}\)