Zero vector, opposite vectors \[ \vec{r} + \vec{r\,}' = 0 \quad \Longleftrightarrow \quad \vec{r\,}' = - \vec{r} \]

Can add vectors in any order \[ \vec{W} + \vec{T} + \vec{F} = \left(\vec{W} + \vec{T}\right) + \vec{F} = \vec{W} + \left(\vec{T} + \vec{F}\right) \] Vector components

• resolve $\vec{T}$ into component vectors $\vec{T}_\mbox{h}$ and $\vec{T}_\mbox{v}$ \[ \vec{T} = \vec{T}_\mbox{h} + \vec{T}_\mbox{v} \]
• unit vectors $\hat{x}$, $\hat{y}$ define the reference frame
• scalar components $r_x$, $r_y$ of vector $\vec{r}=r\angle \theta $ $$ \vec{r} = r_x \> \hat{x} + r_y \> \hat{y} $$
• from $r\angle\theta$ to $r_x$, $r_y$ : measure $\theta$ counterclockwise from $+x$ direction \[ \left\{ \begin{array}{lll} r_x = r \> \cos\theta \\ r_y = r \> \sin\theta \end{array} \right. \]
• from $r_x$, $r_y$ to $r\angle\theta$ : Pythagorean theorem \[ \left\{ \begin{array}{l} r= \sqrt{ \> r_x^2 + r_y^2} \\ \theta = \arctan{{r_y}\over{r_x}} \end{array} \right. \]