7.3: 垂直线
- Page ID
- 171087
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)两条截然不同的直线\(l\)\(l ⊥ q\)，如果它们的交点在测量时形成四个直角或角度，则\(q\)是垂直的\(90^{\circ}\)。 垂直线的斜率\(l\)和\(q\)是负倒数。 也就是说，
\[m_l = −\dfrac{1}{m_q} \nonumber \]
和
\[m_q = − \dfrac{1}{m_l} \nonumber \]
确定给定线是否垂直。 穿\(l\)过点\((0, 1)\)和的直线\((1, 3)\)，以及穿\(q\)过点\((−1, 4)\)和的线\((5, 1)\)。
解决方案
要确定直线是否垂直，首先使用直线公式的斜率求出它们的斜率。 穿过点的直线的\(l\)斜\((0, 1)\)率\((1, 3)\)是\(m_l\)
\(\begin{array}s m_l &= \dfrac{3 − 1}{1 − 0} \\ &= \dfrac{2}{1} \\ &= 2 \end{array}\)
穿过点\((−1, 4)\)的直线的\(q\)斜率为\(m_q\)\((5, 1)\)
\(\begin{array}s m_q &= \dfrac{1 − 4}{5 − (-1)} \\ &= \dfrac{-3}{6} \\ &= \dfrac{-1}{2} \end{array}\)
现在，\(q\)当\(l\)且仅在以下情况下，直线和才是垂直的：
\(m_l = −\dfrac{1}{m_q} \text{ and } m_q = −\dfrac{1}{m_l}\)
\(m_l = 2\)和\(m_q = −\dfrac{1}{m_l} = −\dfrac{1}{2}\)。 因此，直线的斜率是负倒数，因此可以得出结论，直线\(l\)和\(q\)是垂直线。
找出一条垂直于穿过点的直线的斜率\(l\)，\((−3, 0)\)然后\((3, 4)\)。
解决方案
首先找到穿过点的线的\(l\)斜率\((3, 4)\)，\((−3, 0)\)然后使用直线公式的斜率。 因此，
\(\begin{array} s m_l &= \dfrac{y_2 − y_1}{x_2 − x_1} \\ &= \dfrac{4 − 0}{3 − (−3)} \\ &= \dfrac{4}{6} \\ &= \dfrac{2}{3} \end{array}\)
任何垂直于直线的直线的斜率\(l\)必须为其斜率的负倒数。 从\(m_l = \dfrac{2}{3}\)那时起，垂直于直线的线的斜率\(l\)必须为\(m = −\dfrac{3}{2}\)
确定给定线是否垂直。
- 穿\(l\)过点的直线\((0, 4)\)\((5, 3)\)\(q\)和穿过点的直线\((1, 5)\)和\((−1, −5)\)。
- 穿\(l\)过点的直线\((−2, −5)\)\((1, 7)\)\(q\)和穿过点的直线\((−4, 1)\)和\((−3, −3)\)。
找出垂直于以下位置的直线的斜率：
- 穿\(l\)过点\((4, 2)\)和的线\((−1, −2)\)。
- 穿\(q\)过点\((7, −8)\)和的线\((9, 1)\)。