利用向量法解立体几何问题

利用向量法解立体几何问 利用法向量解立体几何题 一、运用法向量求空间角 ,,,,,,,, 向量法求空间两条异面直线a, b所成角θ，只要在两条异面直线a, b上各任取一个向量，则角AABB''和 ,,,,,,,,,,,,,,,,AABB'',<>=θ或π-θ，因为θ是锐角，所以cosθ=, 不需要用法向量。 ,,,,,,,,AABB',' AABB'',1、运用法向量求直线和平面所成角 A , 设平面α的法向量为=(x, y, 1)，则直线AB和平面αn n 所成的角θ的正弦值为 ,,,,,, ,,,,AB,n,sinθ= cos(-θ) = |cos<, >| = n,,,,,AB2AB,nα 2、运用法向量求二面角 ,,,,,,,,,,,,,,,设二面角的两个面的法向量为，则<>或π-<>是所求角。这时要借助图形来判断所求角nn,nn,nn,121212 ,,,,,,,,,, >是所求，还是π-<>是所求角。 为锐角还是钝角，来决定(或其补角) AAn//BAn, ,,,,, ||ABn,,,?异面直线a、b的距离d =AB?cos?BAA= * ||n ,,,,,,,,,,,,n其中，的坐标可利用a、b上的任一向量(或图中的)，及的定义得 nAEBF,ab, ,,,,,,,nana,,,0,,n, ? 解方程组可得。 ,,,,,,nbnb,,,0,,,, geographic location Yibin city is located in beautiful rich of "abundance"--Sichuan basin South margin, is located in n 103 ? 39 '-105 ? 20 ', latitude 27 ? 49 '-29 ? 16 ' Zhijian; Yibin East and, Luzhou city, adjacent; North and zigong city development 12th a five years planning platform for into Yu city inter track network planning ( 2011-2030) national and place of related planning, and policy Yibin City tourism development planning Yibin city ease city hold blocking special planning Yibin city related tablets district control sex detailed planning Yibin city public traffic development planning Yibin city land using general planning Yibin city industrial concentrated district layout planning 1.4 planning background Yibin city in sands River, and minjiang River and Yangtze River of Sanjiang intersection at, is China famous of "wine are", belongs to chuan, and Yunnan, and Guizhou three province combined Department area, known as " Yangtze River City, "said, is" in panxi-liupanshui "important part of the resource-rich region. In the new master plan, as the national historical and cultural city in Yibin city, China, Department of Sichuan-Yunnan-Guizhou-based regional center in the upper reaches Counties are: Yibin, and Jiang ' an, and changning County, and Gao Pingshan County, xingwen junlian, and Gong, and, and, and. 2.1-2 administrative map of Yibin, Yibin city, end of 2014 the city's total population of 5.5429 million people, resident population of 825,000 people in the Center. 2.1.3 social economic development in Yibin city, 2014 to achieve GDP of 144.381 billion yuan, an increase of 8%, keeping the total province-4th place; local public budget income of 10.56 billion yuan, ranking 4th place, above-scale industrial added value of 78.716 billion yuan, ranking 3rd in social retail sales of consumer goods totaled 56.169 billion yuan, ranking 4th in 24990 Yuan of disposable income per capita of urban residents, per capita net income of farmers 9831, Share 123 14.5:60.0:25.5. Wine-making industry, 2、求点到面的距离 ,求A点到平面α的距离，设平面α的法向量法为，在α内任取一点B，则A点到平面α的距nxy,(,,1) ,,,,,,,||ABn,离为d =，的坐标由与平面α内的两个不共线向量的垂直关系，得到方程组(类似于前面所述, 若,nn ||n , 方程组无解，则法向量与XOY平面平行，此时可改设，下同)。 ny,(1,,0)3、求直线到与直线平行的平面的距离 ,求直线a到平面α的距离，设平面α的法向量法为，在直线a上任取一点A，在平面α内任取nxy,(,,1) ,,,,, ||ABn,一点B，则直线a到平面α的距离d = , ||n 4、求两平行平面的距离 , 设两个平行设平面α、β的公共法向量法为，在平面α、β内各任取一点A、B，则平面α到nxy,(,,1) ,,,,, ||ABn,平面β的距离d = , ||n 三、证明线面、面面的平行、垂直关系 ,,,,,设平面外的直线a和平面α、β，两个面α、β的法向量为，则 nn,12 ,,,,, a//a,,,naa//n,,,11 ,,,,,,,,,, ,,////,nn,,,,,nn1212 四、应用举例: 例1:(广东18)如右下图,在长方体ABCD—ABCD中,已知AB= 4, AD =3, AA= 2. E、F分别是线段11111 AB、BC上的点，且EB= FB=1. (1) 求二面角C—DE—C的正切值; 1 (2) 求直线EC与FD所成的余弦值. 11,,,,,,,,,,,解:(I)以A为原点，分别为x轴，y轴,z轴的正向建立空间直角坐标系， ABADAA,,1 则D(0,3,0)、D(0,3,2)、E(3,0,0)、F(4,1,0)、C(4,3,2) 11,,,,,,,,,,,, 于是， DEECFD,,,,,(3,3,0),(1,3,2),(4,2,2) 11, 设法向量与平面CDE垂直，则有 nxy,(,,2)1,,,,,,nDE,330xy,,,,,,,,,xy1 ,,,,,,,xyz，，,320,nEC,,,1 2making industry,-Wine 25.5.in 24990 Yuan of disposable income per capita of urban residents, per capita net income of farmers 9831, Share 123 14.5:60.0: d value of 78.716 billion yuan, ranking 3rd in social retail sales of consumer goods totaled 56.169 billion yuan, ranking 4thscale industrial adde-4th place; local public budget income of 10.56 billion yuan, ranking 4th place, above-ing the total provinceCenter. 2.1.3 social economic development in Yibin city, 2014 to achieve GDP of 144.381 billion yuan, an increase of 8%, keep ion of 825,000 people in the2 administrative map of Yibin, Yibin city, end of 2014 the city's total population of 5.5429 million people, resident populat-ang ' an, and changning County, and Gao Pingshan County, xingwen junlian, and Gong, and, and, and. 2.1based regional center in the upper reaches Counties are: Yibin, and Ji-Guizhou-Yunnan-rich region. In the new master plan, as the national historical and cultural city in Yibin city, China, Department of Sichuan-liupanshui "important part of the resource-panxiongs to chuan, and Yunnan, and Guizhou three province combined Department area, known as " Yangtze River City, "said, is" in bin city in sands River, and minjiang River and Yangtze River of Sanjiang intersection at, is China famous of "wine are", belnd Yining Yibin city land using general planning Yibin city industrial concentrated district layout planning 1.4 planning backgrouspecial planning Yibin city related tablets district control sex detailed planning Yibin city public traffic development plan ocking2030) national and place of related planning, and policy Yibin City tourism development planning Yibin city ease city hold bl-; North and zigong city development 12th a five years planning platform for into Yu city inter track network planning ( 201129 ? 16 ' Zhijian; Yibin East and, Luzhou city, adjacent-105 ? 20 ', latitude 27 ? 49 '-Sichuan basin South margin, is located in n 103 ? 39 '--geographic location Yibin city is located in beautiful rich of "abundance" , ？,,,n(1,1,2), ,,,,向量与平面垂直？AACDE,(0,0,2),1 ,,,,,,与所成的角为二面角的平面角 ？,,nAACDEC11 ,,,,, nAA,,，,，，，10102261,？cos,,,,,,,,,, 3||||114004nAA，，，，，，1 2,？,tan 2(II)设EC与FD所成角为β，则 11,,,,,,,, ,ECFD1(4)322221，,，，，，11,,,,,,,,,, cos,,,22222214||||ECFD，132(4)22，，，,，，11 0例2:(辽宁卷17)如图，已知四棱锥P-ABCD，底面ABCD是菱形，?DAB=60，PD?平面ABCD，PD=AD， 点E为AB中点，点F为PD中点。 (1)证明平面PED?平面PAB; (2)求二面角P-AB-F的平面角的余弦值 0证明:(1)?面ABCD是菱形，?DAB=60， ??ABD是等边三角形，又E是AB中点，连结BD 000 ??EDB=30，?BDC=60，??EDC=90， 13 如图建立坐标系D-ECP，设AD=AB=1，则PF=FD=，ED=， 22 ,,,,,,,,113333?P(0，0，1)，E(，0，0)，B(，，0) ?=(，，-1)，= (，0，-1)， PBPE222222 ,,,,, DCn平面PED的一个法向量为=(0，1，0) ，设平面PAB的法向量为=(x, y, 1) ,,3131,,,,,2,(,,1)(,,1)010xyxy,,,,,,,,,,x,nPB,2,,,,2222n由 ?=(, 0, 1) ,,,,,,,,3,,,,3nPE,33,,,,,y,0(,,1)(,0,1)010xyx,,,,,,,,,,22 ,,,,,,,,,, DCDCnn??=0 即? ?平面PED?平面PAB ,,2nn(2)解:由(1)知:平面PAB的法向量为=(, 0, 1), 设平面FAB的法向量为=(x, y, -1)， 13 ,,,,,,,111133FBFE由(1)知:F(0，0，)，=(，，-)， = (，0，-)， 222222 3making industry,-e5. Win24990 Yuan of disposable income per capita of urban residents, per capita net income of farmers 9831, Share 123 14.5:60.0:25.lue of 78.716 billion yuan, ranking 3rd in social retail sales of consumer goods totaled 56.169 billion yuan, ranking 4th in scale industrial added va-4th place; local public budget income of 10.56 billion yuan, ranking 4th place, above-the total province ter. 2.1.3 social economic development in Yibin city, 2014 to achieve GDP of 144.381 billion yuan, an increase of 8%, keepingion of 825,000 people in the Cen2 administrative map of Yibin, Yibin city, end of 2014 the city's total population of 5.5429 million people, resident populat-' an, and changning County, and Gao Pingshan County, xingwen junlian, and Gong, and, and, and. 2.1based regional center in the upper reaches Counties are: Yibin, and Jiang -Guizhou-Yunnan-rich region. In the new master plan, as the national historical and cultural city in Yibin city, China, Department of Sichuan-anshui "important part of the resourceliup-to chuan, and Yunnan, and Guizhou three province combined Department area, known as " Yangtze River City, "said, is" in panxi city in sands River, and minjiang River and Yangtze River of Sanjiang intersection at, is China famous of "wine are", belongsYibin g Yibin city land using general planning Yibin city industrial concentrated district layout planning 1.4 planning background cial planning Yibin city related tablets district control sex detailed planning Yibin city public traffic development planninocking spe2030) national and place of related planning, and policy Yibin City tourism development planning Yibin city ease city hold bl-rth and zigong city development 12th a five years planning platform for into Yu city inter track network planning ( 201129 ? 16 ' Zhijian; Yibin East and, Luzhou city, adjacent; No-105 ? 20 ', latitude 27 ? 49 '-Sichuan basin South margin, is located in n 103 ? 39 '--geographic location Yibin city is located in beautiful rich of "abundance" ,,311311,,,,,1,(,,1)(,,)00xyxy,,,,,，,,,,,x,,nFB,11,,,,222222由 ?=(-, 0, -1) n,,,1,,,,,3,,,,3nFE,31311,,,,,y,0(,,1)(,0,)00xyx,,,,，,,,,,,2222 ,,,,n57,n1, >| = ?二面角P-AB-F的平面角的余弦值cosθ= |cos<,nn,,114n,n1 例3:(江苏高考18)在棱长为4的正方体ABCD-ABCD中，O是正方形ABCD的中心，点P在棱CC上，111111111 且CC=4CP. 1 (?)求直线AP与平面BCCB所成的角的大小(结果用反三角函数值

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示); 11 (?)求点P到平面ABD的距离. 1 解: (?)如图建立坐标系D-ACD, ?棱长为4 ?A(4，0，0)，B(4，4，0)，P(0，4，1) 1 ,,,,,,,, ? = (-4, 4, 1) , 显然=(0，4，0)为平面BCCB的一个法向量， DCAP11 ,,,,,,,,16433 ?直线AP与平面BCCB所成的角θ的正弦值sinθ= |cos<, >|= DCAP,11222334414，，, 433 ?θ为锐角，?直线AP与平面BCCB所成的角θ为arcsin1133 , (?) 设平面ABD的法向量为n=(x, y, 1)， 1 ,,,,,,,,, ?AB=(0，4，0)，=(-4，0，4) AD1 ,,,,,,,,,,,,y,0,nnnAB 由?，? 得 ? =(1, 0, 1)， AD,1,，,440x, ,,,,, APn,32, ?点P到平面ABD的距离 d = , 12n 例4:在长、宽、高分别为2，2，3的长方体ABCD-ABCD中，O是底面中心，求AO与BC的距离。 111111 解:如图，建立坐标系D-ACD，则O(1，1，0)，A(2，2，3)，C(0，2，0) 11 ,,,,,,,,,,,,, ?AO,,,(1,1,3) BC,,,(2,0,3) AB,(0,2,0) 1111 , 设AO与BC的公共法向量为，则 nxy,(,,1)11 4making industry,-Wine 25.5.in 24990 Yuan of disposable income per capita of urban residents, per capita net income of farmers 9831, Share 123 14.5:60.0: d value of 78.716 billion yuan, ranking 3rd in social retail sales of consumer goods totaled 56.169 billion yuan, ranking 4thscale industrial adde-4th place; local public budget income of 10.56 billion yuan, ranking 4th place, above-ing the total provinceCenter. 2.1.3 social economic development in Yibin city, 2014 to achieve GDP of 144.381 billion yuan, an increase of 8%, keep ion of 825,000 people in the2 administrative map of Yibin, Yibin city, end of 2014 the city's total population of 5.5429 million people, resident populat-ang ' an, and changning County, and Gao Pingshan County, xingwen junlian, and Gong, and, and, and. 2.1based regional center in the upper reaches Counties are: Yibin, and Ji-Guizhou-Yunnan-rich region. In the new master plan, as the national historical and cultural city in Yibin city, China, Department of Sichuan-liupanshui "important part of the resource-panxiongs to chuan, and Yunnan, and Guizhou three province combined Department area, known as " Yangtze River City, "said, is" in bin city in sands River, and minjiang River and Yangtze River of Sanjiang intersection at, is China famous of "wine are", belnd Yining Yibin city land using general planning Yibin city industrial concentrated district layout planning 1.4 planning backgrouspecial planning Yibin city related tablets district control sex detailed planning Yibin city public traffic development plan ocking2030) national and place of related planning, and policy Yibin City tourism development planning Yibin city ease city hold bl-; North and zigong city development 12th a five years planning platform for into Yu city inter track network planning ( 201129 ? 16 ' Zhijian; Yibin East and, Luzhou city, adjacent-105 ? 20 ', latitude 27 ? 49 '-Sichuan basin South margin, is located in n 103 ? 39 '--geographic location Yibin city is located in beautiful rich of "abundance" 3,,,,,,x,,,,nAO,(,,1)(1,1,3)030xyxy,,,,,，,,,,,,12,,,,,,,,,,,,(,,1)(2,0,3)02303xyx,,,,,,,nBC,,,,,1,y, ,,2CD1 1 ,33 ? n,,(,,1)AB221 1 ? AO与BC的距离为 11 33,,,,,,,,DC0,2,0,,1,,，，,,22||3322ABn,,,11,,,, d = 22O 11||11n33,,,,AB,，，1,,,,222,,,, 例5:在棱长为1的正方体ABCD-ABCD中，E、F分别是BC、CD的中点，求A到面BDFE的距离。 111111111 1解:如图，建立坐标系D-ACD，则B(1，1，0)，A(1，0，1)，E(，1，1) 112 ,,,,,,,,,,,, 1FC1 ? BE,,(,0,1)AB,,(0,1,1)BD,,,(1,1,0)D1 12 ,EAB1 1 设面BDFE的法向量为，则 nxy,(,,1) ,,,,,(,,1)(1,1,0)00xyxy,,,,,,,,,,nBDx,,2,,,,,,, ,,,,,,,,,11DCy,,2(,,1)(,0,1)010xyx,,,,，,nBE,,,,,,,,22 ,AB ? n,,(2,2,1) ,,,,,0,1,12,2,1,,,，，，，||ABn,|3|,1,? A到面BDFE的距离为d = ,,,11223||n221，,，，， 五、练习: 1、(全国高考题)如图,已知正四棱柱ABCD-ABCD, AB=1,AA=2,点E为CC中点,点F为BD中点. (1) 证1111111明EF为BD与CC的公垂线; (2)求点D到面BDE的距离. 111 2、已知正方形ABCD，边长为1，过D作PD?平面ABCD，且PD=1，E、F分别是AB和BC的中点， (1)求D到平面PEF的距离; (2)求直线AC到平面PEF的距离 D13、在长方体ABCD-ABCD中，AB=4，BC=3，CC=2(如图) C111111 (1)求证:平面ABC//平面ACD; 111 A1B1 (2)求(1)中两个平行平面间的距离; DC (3)求点B到平面ABC的距离。 111 AB2004/7/19 5making industry,-e5. Win24990 Yuan of disposable income per capita of urban residents, per capita net income of farmers 9831, Share 123 14.5:60.0:25.lue of 78.716 billion yuan, ranking 3rd in social retail sales of consumer goods totaled 56.169 billion yuan, ranking 4th in scale industrial added va-4th place; local public budget income of 10.56 billion yuan, ranking 4th place, above-the total province ter. 2.1.3 social economic development in Yibin city, 2014 to achieve GDP of 144.381 billion yuan, an increase of 8%, keepingion of 825,000 people in the Cen2 administrative map of Yibin, Yibin city, end of 2014 the city's total population of 5.5429 million people, resident populat-' an, and changning County, and Gao Pingshan County, xingwen junlian, and Gong, and, and, and. 2.1based regional center in the upper reaches Counties are: Yibin, and Jiang -Guizhou-Yunnan-rich region. In the new master plan, as the national historical and cultural city in Yibin city, China, Department of Sichuan-anshui "important part of the resourceliup-to chuan, and Yunnan, and Guizhou three province combined Department area, known as " Yangtze River City, "said, is" in panxi city in sands River, and minjiang River and Yangtze River of Sanjiang intersection at, is China famous of "wine are", belongsYibin g Yibin city land using general planning Yibin city industrial concentrated district layout planning 1.4 planning background cial planning Yibin city related tablets district control sex detailed planning Yibin city public traffic development planninocking spe2030) national and place of related planning, and policy Yibin City tourism development planning Yibin city ease city hold bl-rth and zigong city development 12th a five years planning platform for into Yu city inter track network planning ( 201129 ? 16 ' Zhijian; Yibin East and, Luzhou city, adjacent; No-105 ? 20 ', latitude 27 ? 49 '-Sichuan basin South margin, is located in n 103 ? 39 '--geographic location Yibin city is located in beautiful rich of "abundance"