2.3: 距离公式
- Page ID
- 171180
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)上一节讲述了如何在矩形坐标平面上绘制点。 本节介绍如何找到平面中任意两点之间的距离。 例如，要计算点的距离\((x_1, y_1)\)，\((x_2, y_2)\)请考虑以下公式：
两点之间\(P_1(x_1, y_1)\)以及平面\(P_2(x_2, y_2)\)中的距离 d 由下式给出：
\(d = \sqrt {(x_2 − x_1) ^2 + (y_2 − y_1)} ^2\)
找出两点\((−5, 2)\)之间的距离\((3, 4)\)
解决方案
让\(P_1(−5, 2)\) an\(P_2(3, 4)\) d 成为平面中的两个点，然后 let\(x_1 = −5\)\(y_1 = 2\)、\(x_2 = 3\)、和\(y_2 = 4\)。
使用给定值的距离公式：
\(\begin{aligned} d &= \sqrt{(x_2 − x_1) ^2 + (y_2 − y_1) ^2 } \\&= \sqrt{ (3 − (−5))^2 + (4 − 2)^2}\\& = \sqrt{ (3 + 5)^2 + (2)^2 } \\ &= \sqrt{ 8 ^2 + 2^2} \\ &= \sqrt{64 + 4 }\\ &= \sqrt{ 68 } \\&= 2\sqrt{17}\end{aligned}\)
因此，两个给定点之间的距离为\(2\sqrt{17}\)。
找出两点\((−2.5, −1)\)和之间的距离\((−3, −1.5)\)。
解决方案
让\(P_1(−2.5, −1)\)和\(P_2(−3, −1.5)\)成为飞机中的点然后 let\(x_1 = −2.5\)\(y_1 = −1\)、\(x_2 = −3\)和\(y_2 = −1.5\)。
然后使用给定值的距离公式得出，
\(\begin{aligned} d &= \sqrt{(x_2 − x_1) ^2 + (y_2 − y_1) ^2}\\& = \sqrt{[−3 − (−2.5)]^2 + [−1.5 − (−1)]^2 } \\&= \sqrt{ (−3 + 2.5)^2 + (−1.5 + 1)^2} \\&= \sqrt{ (−0.5)^2 + (−0.5)^2 } \\&= \sqrt{ 0.25 + 0.25 }\\ &= \sqrt{0.5 } \\&\approx 0.71 \end{aligned}\)
因此，两个给定点之间的距离约为 0.71。
- 找出\(P_1(−3, −1.5)\)和之间的距离\(P_2(−2.5, − 1)\)。 将答案与示例 2 中的答案进行比较。 可以得出什么结论？
- 找出\((−3, 6)\)和之间的距离\((2, 4)\)
- 找出两点\(\left( \dfrac{1 }{2} , − \dfrac{10 }{4}\right)\)之间的距离\(\left(− \dfrac{14 }{4} , − \dfrac{5 }{2}\right )\)
- 为什么要使用距离公式？