18.5: Argument and polar coordinates
- Page ID
- 23700
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)As before, we assume that \(O\) and \(E\) are the points with complex coordinates \(0\) and \(1\) respectively.
Let \(Z\) be a point distinct form \(O\). Set \(\rho=OZ\) and \(\theta=\measuredangle EOZ\). The pair \((\rho,\theta)\) is called the polar coordinates of \(Z\).
If \(z\) is the complex coordinate of \(Z\), then \(\rho=|z|\). The value \(\theta\) is called the argument of \(z\) (briefly, \(\theta=\arg z\)). In this case,
\(z=\rho\cdot e^{i\cdot\theta}=\rho\cdot(\cos\theta+i\cdot\sin\theta).\)
Note that
\(\arg (z\cdot w) \equiv \arg z+\arg w\)
and
\(\arg \tfrac z w \equiv \arg z-\arg w\)
if \(z \ne 0\) and \(w \ne 0\). In particular, if \(Z\), \(V\), \(W\) are points with complex coordinates \(z\), \(v\), and \(w\) respectively, then
\[\begin{aligned} \measuredangle VZW &=\arg\left(\frac{w-z}{v-z}\right)\equiv \\ &\equiv \arg(w-z)-\arg(v-z) \end{aligned}\]
if \(\measuredangle VZW\) is defined.
Use the formula 18.5.1 to show that
\(\measuredangle ZVW+\measuredangle VWZ+\measuredangle WZV\equiv \pi\)
for any \(\triangle ZVW\) in the Euclidean plane.
- Hint
-
Let \(z, v\), and \(w\) denote the complex coordinates of \(Z, V\), and \(W\) respectively. Then
\(\begin{array} {rcl} {\measuredangle ZVW + \measuredangle VWZ + \measuredangle WZV} & \equiv & {\arg \dfrac{w - v}{z- v} + \arg \dfrac{z-w}{v-w} + \arg \dfrac{v-z}{w-z} \equiv} \\ {} & \equiv & {\arg \dfrac{(w - v) \cdot (z - w) \cdot (v -z)}{(z - v) \cdot (v - w) \cdot (w - z)} \equiv} \\ {} & \equiv & {\arg (-1) \equiv \pi} \end{array}\)
Suppose that points \(O\), \(E\), \(V\), \(W\), and \(Z\) have complex coordinates \(0\), \(1\), \(v\), \(w\), and \(z=v\cdot w\) respectively. Show that
\(\triangle OEV\sim \triangle OWZ.\)
- Hint
-
Note and use that \(\measuredangle EOV = \measuredangle WOZ = \arg v\) and \(\dfrac{OW}{OZ} = \dfrac{OZ}{OW} = |v|\).
The following theorem is a reformulation of Corollary 9.3.2 which uses complex coordinates.
Let \(\square UVWZ\) be a quadrangle and \(u\), \(v\), \(w\), and \(z\) be the complex coordinates of its vertices. Then \(\square UVWZ\) is inscribed if and only if the number
\(\dfrac{(v-u)\cdot(z-w)}{(v-w)\cdot(z-u)}\)
is real.
The value \(\dfrac{(v-u)\cdot(w-z)}{(v-w)\cdot(z-u)}\) is called the complex cross-ratio of \(u\), \(w\), \(v\), and \(z\); it will be denoted by \((u,w;v,z)\).
Observe that the complex number \(z\ne 0\) is real if and only if \(\arg z=0\) or \(\pi\); in other words, \(2\cdot\arg z\equiv 0\).
Use this observation to show that Theorem \(\PageIndex{1}\) is indeed a reformulation of Corollary 9.3.2.
- Hint
-
Note that
\(\arg \dfrac{(v-u) \cdot (z-w)}{(v -w) \cdot (z -u)} \equiv \arg \dfrac{v - u}{z - u} + \arg \dfrac{z- w}{v -w} = \measuredangle ZUV + \measuredangle VWZ.\)
The statement follows since the value \(\dfrac{(v - u) \cdot (x - w)}{(v - w) \cdot (z - u)}\) is real if and only if
\(2 \cdot \arg \dfrac{(v - u) \cdot (z - w)}{(v - w) \cdot (z - u)} \equiv 0.\)