7.3: Linhas perpendiculares
- Page ID
- 170043
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Duas linhas distintas\(l\) e\(q\) são perpendiculares, escritas\(l ⊥ q\), se sua interseção formar quatro ângulos retos ou ângulos com medida\(90^{\circ}\). As inclinações das retas perpendiculares\(l\) e\(q\) são recíprocos negativos. Ou seja,
\[m_l = −\dfrac{1}{m_q} \nonumber \]
e
\[m_q = − \dfrac{1}{m_l} \nonumber \]
Determine se as linhas fornecidas são perpendiculares. A linha\(l\) que passa pelos pontos\((0, 1)\) e\((1, 3)\), e a linha\(q\) que passa pelos pontos\((−1, 4)\)\((5, 1)\) e.
Solução
Para determinar se as linhas são perpendiculares, primeiro encontre suas inclinações usando a fórmula da inclinação da linha. A inclinação da linha\(l\)\(m_l\), que passa pelos pontos\((0, 1)\) e\((1, 3)\) é,
\(\begin{array}s m_l &= \dfrac{3 − 1}{1 − 0} \\ &= \dfrac{2}{1} \\ &= 2 \end{array}\)
A inclinação da linha\(q\),\(m_q\), que passa pelos pontos\((−1, 4)\) e\((5, 1)\), é
\(\begin{array}s m_q &= \dfrac{1 − 4}{5 − (-1)} \\ &= \dfrac{-3}{6} \\ &= \dfrac{-1}{2} \end{array}\)
Agora, as linhas\(l\) e\(q\) são perpendiculares se e somente se:
\(m_l = −\dfrac{1}{m_q} \text{ and } m_q = −\dfrac{1}{m_l}\)
\(m_l = 2\)\(m_q = −\dfrac{1}{m_l} = −\dfrac{1}{2}\)e. Portanto, as inclinações das retas são recíprocas negativas, então pode-se concluir que as retas\(l\) e\(q\) são retas perpendiculares.
Encontre a inclinação de uma linha perpendicular à linha\(l\) que passa pelos pontos\((−3, 0)\)\((3, 4)\) e.
Solução
Comece encontrando a inclinação da linha\(l\) que passa pelos pontos\((−3, 0)\) e\((3, 4)\), usando a fórmula da inclinação da linha. Assim,
\(\begin{array} s m_l &= \dfrac{y_2 − y_1}{x_2 − x_1} \\ &= \dfrac{4 − 0}{3 − (−3)} \\ &= \dfrac{4}{6} \\ &= \dfrac{2}{3} \end{array}\)
Qualquer linha perpendicular à linha\(l\) deve ter uma inclinação negativa recíproca de sua inclinação. Desde\(m_l = \dfrac{2}{3}\) então, a inclinação da linha perpendicular à linha\(l\) deve ser\(m = −\dfrac{3}{2}\)
Determine se as linhas fornecidas são perpendiculares.
- A linha\(l\) que passa pelos pontos\((0, 4)\)\((5, 3)\) e a linha\(q\) que passa pelos pontos\((1, 5)\)\((−1, −5)\) e.
- A linha\(l\) que passa pelos pontos\((−2, −5)\)\((1, 7)\) e a linha\(q\) que passa pelos pontos\((−4, 1)\)\((−3, −3)\) e.
Encontre a inclinação de uma linha perpendicular a:
- Linha\(l\) que passa pelos pontos\((4, 2)\)\((−1, −2)\) e.
- Linha\(q\) que passa pelos pontos\((7, −8)\)\((9, 1)\) e.