Home > Courses > 1P21_Sternin > Kinematics Algebra of vectors 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 $\left\{ \begin{array}{l} r= \sqrt{ \> r_x^2 + r_y^2} \\ \theta = \arctan{{r_y}\over{r_x}} \end{array} \right.$ Relating vectors and their components two vectors are equal, $\vec{r} = \vec{r\,}'$ $$\Updownarrow$$ their magnitudes are equal, $r = r'$ and their directions are the same, $\theta = \theta '$ $$\Updownarrow$$ their components are equal: $\left\{ \begin{array}{lll} r_x = r \> \cos\theta = r'_x = r'\> \cos\theta' \\ r_y = r \> \sin\theta = r'_y = r'\> \sin\theta' \end{array} \right.$