Direction Cosines

$$\begin{array}{r}\cos \alpha =\frac{u_1}{\Vert \vec{u}\Vert }=\frac{\vec{u}\cdot {\hat{e}}_1}{\Vert \vec{u}\Vert }\\ \cos \beta =\frac{u_2}{\Vert \vec{u}\Vert }=\frac{\vec{u}\cdot {\hat{e}}_2}{\Vert \vec{u}\Vert }\\ \cos \gamma =\frac{u_3}{\Vert \vec{u}\Vert }=\frac{\vec{u}\cdot {\hat{e}}_3}{\Vert \vec{u}\Vert }\end{array}$$

Properties

• ${\cos }^2\alpha +{\cos }^2\beta +{\cos }^2\gamma =1$
• $\hat{u}=⟨\cos \alpha ,\cos \beta ,\cos \gamma ⟩$