7.2: الخطوط المتوازية
- Page ID
- 166944
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)في المستوى الإحداثي، الخطوط المتوازية هي خطوط لا تلتقي أو تتقاطع. إنهم دائمًا على مسافة واحدة. علاوة على ذلك، فإن الخطوط المتوازية لها نفس المنحدر.
أوجد ميل الخط\(l\) الذي يمر عبره\((2, 0)\)\((4, −3)\) ومنحدر الخط\(q\) الذي يمر عبره\((2, −3)\) و\((4, −6)\). حدد ما إذا كانت الخطوط متوازية.
الحل
استخدم ميل صيغة الخط المستقيم للعثور على ميل الخط\(l\)\(m_l\)، ومنحدر الخط\(q\)\(m_q\)، كما يلي،
\(\begin{array} &&m_l = \dfrac{y_2 − y_1}{x_2 − x_1}\;\;\;\;\;\;\;\;\;\; &m_q = \dfrac{y_2 − y_1}{x_2 − x_1} \\ &= \dfrac{−3 − 0}{4 − 2}\;\;\;\;\;\;\;\;\;\; &= \dfrac{−6 − (−3)}{4 − 2} \\ &= \dfrac{−3}{2}\;\;\;\;\;\;\;\;\;\; &= \dfrac{−3}{2} \end{array}\)
وبما أن المنحدرين متساويان، فإن الخطوط\(l\)\(q\) متوازية.
حدِّد ما إذا كانت الخطوط المُعطاة متوازية أم لا:
- الخط\(l\) الذي يمر عبر النقاط\((2, 2)\)\((3, 3)\) والخط\(q\) الذي يمر عبر النقاط\((4, 1)\) و\((0, 5)\).
- الخط\(l\) الذي يمر عبر النقاط\((1, 3)\)\((6, −2)\) والخط\(q\) الذي يمر عبر النقاط\((−2, −7)\) و\((10, 5)\).
- الخط\(l\) الذي يمر عبر النقاط\((−6, 5)\)\((2, −1)\) والخط\(q\) الذي يمر عبر النقاط\((−4, 0)\) و\((0, −3)\).