直線方程
平面方程
一般式
適用於所有直線的方程:
![直線[數學概念]](/img/8/76f/wZwpmLzUTN1kjM1MzNwIDN0UTMyITNykTO0EDMwAjMwUzLzczL1EzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
![直線[數學概念]](/img/2/e11/wZwpmL2czNwATM4ADMwADN0UTMyITNykTO0EDMwAjMwUzLwAzLyMzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
![直線[數學概念]](/img/9/b7b/wZwpmLxUTM4kDNykzN5ADN0UTMyITNykTO0EDMwAjMwUzL5czL0YzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
(其中 、 不能同時為0)
點斜式
![直線[數學概念]](/img/8/dd4/wZwpmL3MzNxgDOyUjNxIDN0UTMyITNykTO0EDMwAjMwUzL1YzLwAzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
![直線[數學概念]](/img/7/79d/wZwpmLyMDO1UDMxQzMxADN0UTMyITNykTO0EDMwAjMwUzL0MzL2QzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
知道直線上一點 ,並且直線的斜率 存在,則直線可表示為:
![直線[數學概念]](/img/2/a02/wZwpmLxgjM1kjMwkDO0ATN0UTMyITNykTO0EDMwAjMwUzL5gzL2AzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
![直線[數學概念]](/img/7/79d/wZwpmLyMDO1UDMxQzMxADN0UTMyITNykTO0EDMwAjMwUzL0MzL2QzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
當 不存在時,直線可表示為:
![直線[數學概念]](/img/c/9b1/wZwpmLwUTOxAjN1EDNxMDN0UTMyITNykTO0EDMwAjMwUzLxQzL3EzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
斜截式
![直線[數學概念]](/img/2/047/wZwpmL4YzMwUTN3ADO3EDN0UTMyITNykTO0EDMwAjMwUzLwgzLygzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
![直線[數學概念]](/img/9/b7b/wZwpmLxUTM4kDNykzN5ADN0UTMyITNykTO0EDMwAjMwUzL5czL0YzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
![直線[數學概念]](/img/b/365/wZwpmLzcDM1QjMyUjNxIDN0UTMyITNykTO0EDMwAjMwUzL1YzL1MzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
![直線[數學概念]](/img/7/79d/wZwpmLyMDO1UDMxQzMxADN0UTMyITNykTO0EDMwAjMwUzL0MzL2QzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
知道直線在 軸上截距為 (即經過點 ),斜率為 ,直線可表示為:
![直線[數學概念]](/img/b/e40/wZwpmLwUDM1AjMwMzNxIDN0UTMyITNykTO0EDMwAjMwUzLzczL1IzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
![直線[數學概念]](/img/7/79d/wZwpmLyMDO1UDMxQzMxADN0UTMyITNykTO0EDMwAjMwUzL0MzL2QzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
當 不存在時,直線可表示為:
![直線[數學概念]](/img/c/9b1/wZwpmLwUTOxAjN1EDNxMDN0UTMyITNykTO0EDMwAjMwUzLxQzL3EzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
截距式
![直線[數學概念]](/img/1/036/wZwpmL3QzM5YjNzMjM0EDN0UTMyITNykTO0EDMwAjMwUzLzIzL3EzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
![直線[數學概念]](/img/2/fe8/wZwpmL3UTNxEzM1EDNxMDN0UTMyITNykTO0EDMwAjMwUzLxQzL4IzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
![直線[數學概念]](/img/2/047/wZwpmL4YzMwUTN3ADO3EDN0UTMyITNykTO0EDMwAjMwUzLwgzLygzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
![直線[數學概念]](/img/b/365/wZwpmLzcDM1QjMyUjNxIDN0UTMyITNykTO0EDMwAjMwUzL1YzL1MzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
知道直線與 軸交於 ,與 軸交於 ,則直線可表示為:
![直線[數學概念]](/img/c/eac/wZwpmL2UDOwMzN1ADO3EDN0UTMyITNykTO0EDMwAjMwUzLwgzL2EzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
![直線[數學概念]](/img/2/e11/wZwpmL2czNwATM4ADMwADN0UTMyITNykTO0EDMwAjMwUzLwAzLyMzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
![直線[數學概念]](/img/9/b7b/wZwpmLxUTM4kDNykzN5ADN0UTMyITNykTO0EDMwAjMwUzL5czL0YzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
![直線[數學概念]](/img/6/814/wZwpmL4EzN0EDM1MjN1ATN0UTMyITNykTO0EDMwAjMwUzLzYzLyczLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
當 、 均不為0時,斜截式可寫為
該表達式不適用於和任意坐標軸垂直的直線
兩點式
![直線[數學概念]](/img/8/dd4/wZwpmL3MzNxgDOyUjNxIDN0UTMyITNykTO0EDMwAjMwUzL1YzLwAzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
![直線[數學概念]](/img/c/155/wZwpmLzEzMwETO5gDM5IDN0UTMyITNykTO0EDMwAjMwUzL4AzL0IzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
知道直線經過點 和點 ,且斜率存在,則直線可表示為:
![直線[數學概念]](/img/4/f55/wZwpmLxgTO4QjMwkDO0ATN0UTMyITNykTO0EDMwAjMwUzL5gzLzUzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
法線式
![直線[數學概念]](/img/0/c6d/wZwpmL4QTM3QTOwMjM4IDN0UTMyITNykTO0EDMwAjMwUzLzIzLwYzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
![直線[數學概念]](/img/5/fa3/wZwpmL2ADOzEDMwUTMwEDN0UTMyITNykTO0EDMwAjMwUzL1EzL1gzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
![直線[數學概念]](/img/4/055/wZwpmL3gzNzYTO1cjN1IDN0UTMyITNykTO0EDMwAjMwUzL3YzL2YzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
![直線[數學概念]](/img/1/036/wZwpmL3QzM5YjNzMjM0EDN0UTMyITNykTO0EDMwAjMwUzLzIzL3EzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
其中 為原點到直線的距離, 為法線與 軸正方向的夾角
點方向式
![直線[數學概念]](/img/8/dd4/wZwpmL3MzNxgDOyUjNxIDN0UTMyITNykTO0EDMwAjMwUzL1YzLwAzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
![直線[數學概念]](/img/8/7ad/wZwpmLyczM3gjN1MzNwIDN0UTMyITNykTO0EDMwAjMwUzLzczLzMzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
![直線[數學概念]](/img/1/2e5/wZwpmLyYDMzgzMwMzMzIDN0UTMyITNykTO0EDMwAjMwUzLzMzL1IzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
![直線[數學概念]](/img/1/036/wZwpmL3QzM5YjNzMjM0EDN0UTMyITNykTO0EDMwAjMwUzLzIzL3EzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
![直線[數學概念]](/img/2/047/wZwpmL4YzMwUTN3ADO3EDN0UTMyITNykTO0EDMwAjMwUzLwgzLygzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
知道直線上一點 , 、 不等於0,並且直線不與 軸、 軸平行,則直線可表示為:
![直線[數學概念]](/img/6/ea2/wZwpmL2gTOyQjMwMjM4IDN0UTMyITNykTO0EDMwAjMwUzLzIzL1UzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
點法向式
![直線[數學概念]](/img/d/b55/wZwpmLyMDM1kDN2cjN1IDN0UTMyITNykTO0EDMwAjMwUzL3YzLyMzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
空間方程
1. 一般方程 :
![直線[數學概念]](/img/2/07e/wZwpmL1EzM2gzN4kTNwMDN0UTMyITNykTO0EDMwAjMwUzL5UzL3UzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
2. 點向式方程 :
設直線方向向量為(m,n,p ),經過點( x,y,z)
![直線[數學概念]](/img/5/17f/wZwpmL4AzNwQzNyMTMzEDN0UTMyITNykTO0EDMwAjMwUzLzEzL1UzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
3. x0y式
x=kz+b,y=lz+b
有關內容
“角”
設平面e的法向量為c 直線m、n的方向向量為a、b
把平面ax+by+cz+d=0的法向量為(a,b,c);直線x=kz+b,y=lz+a的方向向量為(k,l,1)代入即可
則直線所成的角:m,n所成的角為a。
cosa=cos<a,b>=|a*b|/|a||b|
直線和平面所成的角: 設b為m和e所成的角,則b=π/2±<a,c>。sinb=|cos<a,c>|=|a*c|/|a||c|
平面兩直線所成的角:設K(l)=k,K(l)=k(kk≠-1),tan<l,l>=(k-k)/(1+kk)
距離
異面直線的距離:l、l為異面直線,l,l公垂直線的方向向量為n、C、D為l、l上任意一點,l1到l2的距離為|AB|=|CD*n|/|n|
點到平面的距離:設PA為平面的一條斜線,O是P點在a內的射影,PA和a所成的角為b,n為a的法向量。
易得:|PO|=|PA|sinb=|PA|*|cos<PA,n>|=|PA|*(|PA*n|/|PA||n|)=|PA*n|/|PA|
直線到平面的距離為在直線上一點到平面的距離;
點到直線的距離:A∈l,O是P點在l上的射影,PA和l所成的角為b,s為l的方向向量。
易得:|PO|=|PA|*|sinb|=|PA|*|sin<PA,s>|=|(PA| |s| |-|PA*s| ) /|s|
平面內:直線ax+by+c=0到M(m,n)的距離為|am+bn+c|/(a +b
平行直線:l1:ax+by+c=0,l2:ax+by+d=0,l1到l2的距離為|c-d|/(a +b )
備註:
直線是曲線的暫短停留。
套用
點與直線
一般情況下,點與直線的距離,是指點到直線的最短距離,即垂直距離。
在二維直角坐標中,直線 Ax+By+C=0 與點 (p,q) 的最短距離為
![直線[數學概念]](/img/e/bf9/wZwpmLwIDN5cTN1MzNwIDN0UTMyITNykTO0EDMwAjMwUzLzczLyEzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
![直線[數學概念]](/img/9/864/wZwpmL2gTO3gDM1MjN1ATN0UTMyITNykTO0EDMwAjMwUzLzYzLzQzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
![直線[數學概念]](/img/6/d0e/wZwpmLwcDNzkTO5gDM5IDN0UTMyITNykTO0EDMwAjMwUzL4AzLyEzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
給出向量式 和 點 ,則有距離
![直線[數學概念]](/img/1/798/wZwpmLwgDM4EzN1cjN1IDN0UTMyITNykTO0EDMwAjMwUzL3YzL2MzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
直線相交點
不考慮重合的情形,在二維平面中,兩條相交直線可以相交或平行。
![直線[數學概念]](/img/5/f31/wZwpmL2EzM0YzN1MzNwIDN0UTMyITNykTO0EDMwAjMwUzLzczLzEzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
![直線[數學概念]](/img/9/8f1/wZwpmLxYjM5czN1MzNwIDN0UTMyITNykTO0EDMwAjMwUzLzczL3czLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
給定兩條直線 和 ,二者相交的條件是
![直線[數學概念]](/img/3/2f0/wZwpmL4EjM2QTMxMTMzEDN0UTMyITNykTO0EDMwAjMwUzLzEzLyczLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
。
或等價地,
![直線[數學概念]](/img/6/dc6/wZwpmL1EDN3UjNzMjM0EDN0UTMyITNykTO0EDMwAjMwUzLzIzL0czLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
,
![直線[數學概念]](/img/7/025/wZwpmLyQzNzMzMwMjM4IDN0UTMyITNykTO0EDMwAjMwUzLzIzLzMzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
當中 。
這時兩線的相交點可從克萊姆法則求得
![直線[數學概念]](/img/e/a19/wZwpmLyczM3QTNxMjM4IDN0UTMyITNykTO0EDMwAjMwUzLzIzL2EzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
相交直線夾角
若兩線相交,則會形成夾角。兩線之間的夾角,通常指不大於90°的一隻。
在二維平面上,給定直線y=mx+b,該線與 x-軸的夾角為
![直線[數學概念]](/img/5/45e/wZwpmL0IDM0QzMxgDNxMDN0UTMyITNykTO0EDMwAjMwUzL4QzL2EzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
。
![直線[數學概念]](/img/2/d7b/wZwpmL4YDM2IzMxgTN2IDN0UTMyITNykTO0EDMwAjMwUzL4UzL0UzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
![直線[數學概念]](/img/4/de3/wZwpmL0YDO0MzM0UTM5IDN0UTMyITNykTO0EDMwAjMwUzL1EzL0UzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
給定兩條直線和 ,二者互相垂直若且唯若
![直線[數學概念]](/img/e/dac/wZwpmL1cDN5EjM0UTM5IDN0UTMyITNykTO0EDMwAjMwUzL1EzL0QzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
。
![直線[數學概念]](/img/4/055/wZwpmL3gzNzYTO1cjN1IDN0UTMyITNykTO0EDMwAjMwUzL3YzL2YzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
![直線[數學概念]](/img/1/b0f/wZwpmL0QzMxITN1IzM0IDN0UTMyITNykTO0EDMwAjMwUzLyMzL0czLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
而其他情況,兩線相交所形成的夾角(),則由
![直線[數學概念]](/img/1/e41/wZwpmLwITMwQzM4kTNwMDN0UTMyITNykTO0EDMwAjMwUzL5UzLzYzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
給出。
![直線[數學概念]](/img/5/867/wZwpmLyETMyYDNwMzNxIDN0UTMyITNykTO0EDMwAjMwUzLzczL0YzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
![直線[數學概念]](/img/0/586/wZwpmL3AjM3UjM5ATO4EDN0UTMyITNykTO0EDMwAjMwUzLwkzLygzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
給定相交直線向量式和,則有
![直線[數學概念]](/img/6/bb7/wZwpmL1YDM0EDN1MzNwIDN0UTMyITNykTO0EDMwAjMwUzLzczL0czLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
。
直線的距離
一般情況下,兩條直線的距離,是指最短距離。
二維情況下,兩條相交直線的距離必然為 0 。
![直線[數學概念]](/img/f/ab7/wZwpmL4ETO4cTMxMTMzEDN0UTMyITNykTO0EDMwAjMwUzLzEzL1EzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
![直線[數學概念]](/img/1/9ff/wZwpmLzgzN1UDO1ADO3EDN0UTMyITNykTO0EDMwAjMwUzLwgzLwAzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
若有兩條平行直線及,則有距離
![直線[數學概念]](/img/a/6be/wZwpmLxgDO5ADOxgDNxMDN0UTMyITNykTO0EDMwAjMwUzL4QzLxEzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
。
![直線[數學概念]](/img/e/887/wZwpmLyAzMzYTNzMjM0EDN0UTMyITNykTO0EDMwAjMwUzLzIzLzczLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
![直線[數學概念]](/img/1/2d4/wZwpmLzQDO5ATMxgTN2IDN0UTMyITNykTO0EDMwAjMwUzL4UzL4IzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
給定平行向量式和,則有
![直線[數學概念]](/img/9/8c2/wZwpmLyMjN4kTNwMjM4IDN0UTMyITNykTO0EDMwAjMwUzLzIzL1MzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
。