新高考數(shù)學(xué)一輪復(fù)習(xí)第7章 第09講 立體幾何與空間向量 章節(jié)總結(jié) 講（學(xué)生版）

第09講立體幾何與空間向量章節(jié)總結(jié)(精講）第一部分：典型例題講解題型一：空間位置關(guān)系證明的傳統(tǒng)法與向量法角度1：用傳統(tǒng)法證明空間的平行和垂直關(guān)系角度2：利用向量證明空間的平行和垂直關(guān)系題型二：空間角的向量求法角度1：用傳統(tǒng)法求異面直線所成角角度2：用向量法求異面直線所成角角度3：用向量法解決線面角的問(wèn)題（定值+探索性問(wèn)題（最值，求參數(shù)））角度4：用向量法解決二面角的問(wèn)題（定值+探索性問(wèn)題（最值，求參數(shù)））題型三：距離問(wèn)題角度1：點(diǎn)SKIPIF1<0到直線SKIPIF1<0的距離角度2：點(diǎn)SKIPIF1<0到平面SKIPIF1<0的距離（等體積法）角度3：點(diǎn)SKIPIF1<0到平面SKIPIF1<0的距離（向量法）題型四：立體幾何折疊問(wèn)題第二部分：高考真題感悟第一部分：典型例題剖析第一部分：典型例題剖析題型一：空間位置關(guān)系證明的傳統(tǒng)法與向量法角度1：用傳統(tǒng)法證明空間的平行和垂直關(guān)系典型例題例題1．（2022·四川成都·高一期末（文））如圖，四邊形ABCD為長(zhǎng)方形，SKIPIF1<0，SKIPIF1<0，點(diǎn)SKIPIF1<0、SKIPIF1<0分別為SKIPIF1<0、SKIPIF1<0的中點(diǎn)．設(shè)平面SKIPIF1<0平面SKIPIF1<0．(1)證明：SKIPIF1<0平面SKIPIF1<0；(2)證明：SKIPIF1<0．例題2．（2022·遼寧葫蘆島·高一期末）如圖，在四面體SKIPIF1<0中，SKIPIF1<0，SKIPIF1<0，點(diǎn)SKIPIF1<0是SKIPIF1<0的中點(diǎn)，SKIPIF1<0，且直線SKIPIF1<0面SKIPIF1<0．(1)直線SKIPIF1<0直線SKIPIF1<0；(2)平面SKIPIF1<0平面SKIPIF1<0．例題3．（2022·福建·廈門(mén)市湖濱中學(xué)高一期中）如圖，在正方體SKIPIF1<0中，SKIPIF1<0為SKIPIF1<0的中點(diǎn)，SKIPIF1<0為SKIPIF1<0的中點(diǎn)．(1)求證：SKIPIF1<0平面SKIPIF1<0；(2)求證：平面SKIPIF1<0平面SKIPIF1<0．例題4．（2022·甘肅酒泉·高二期末（文））如圖，在四棱錐SKIPIF1<0中，SKIPIF1<0是邊長(zhǎng)為2的正三角形，SKIPIF1<0，SKIPIF1<0，SKIPIF1<0，SKIPIF1<0，SKIPIF1<0，SKIPIF1<0，SKIPIF1<0分別是線段SKIPIF1<0，SKIPIF1<0的中點(diǎn)．(1)求證：SKIPIF1<0平面SKIPIF1<0；(2)求證：平面SKIPIF1<0平面SKIPIF1<0．角度2：利用向量證明空間的平行和垂直關(guān)系典型例題例題1．（2022·全國(guó)·高二專(zhuān)題練習(xí)）如圖，在直三棱柱SKIPIF1<0中，SKIPIF1<0為SKIPIF1<0的中點(diǎn)．(1)證明：SKIPIF1<0平面SKIPIF1<0；(2)證明：平面SKIPIF1<0平面SKIPIF1<0．例題2．（2022·全國(guó)·高二課時(shí)練習(xí)）如圖所示，在直四棱柱SKIPIF1<0中，底面SKIPIF1<0為等腰梯形，SKIPIF1<0，SKIPIF1<0，SKIPIF1<0，SKIPIF1<0，SKIPIF1<0，SKIPIF1<0，SKIPIF1<0分別是棱SKIPIF1<0，SKIPIF1<0，SKIPIF1<0的中點(diǎn)．求證：（1）直線SKIPIF1<0平面SKIPIF1<0；（2）平面SKIPIF1<0平面SKIPIF1<0．例題3．（2022·全國(guó)·高二專(zhuān)題練習(xí)）如圖，四棱錐SKIPIF1<0中，SKIPIF1<0底面SKIPIF1<0，SKIPIF1<0，SKIPIF1<0，SKIPIF1<0SKIPIF1<0，SKIPIF1<0是SKIPIF1<0的中點(diǎn)．求證：（1）SKIPIF1<0；（2）SKIPIF1<0平面SKIPIF1<0．例題4．（2022·全國(guó)·高三專(zhuān)題練習(xí)）已知正方體SKIPIF1<0中，SKIPIF1<0為棱SKIPIF1<0上的動(dòng)點(diǎn)．（1）求證：SKIPIF1<0；（2）若平面SKIPIF1<0平面SKIPIF1<0，試確定SKIPIF1<0點(diǎn)的位置．題型二：空間角的向量求法角度1：用傳統(tǒng)法求異面直線所成角典型例題例題1．（2022·重慶·西南大學(xué)附中高一期末）正四面體SKIPIF1<0中，SKIPIF1<0，SKIPIF1<0分別是SKIPIF1<0和SKIPIF1<0的中點(diǎn)，則異面直線SKIPIF1<0和SKIPIF1<0所成角的余弦值為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0例題2．（2022·福建莆田·高二期末）若正六棱柱SKIPIF1<0底面邊長(zhǎng)為1，高為SKIPIF1<0，則直線SKIPIF1<0和SKIPIF1<0所成的角大小為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0例題3．（2022·河北邯鄲·高一期末）如圖，在圓臺(tái)SKIPIF1<0中，SKIPIF1<0，SKIPIF1<0，且SKIPIF1<0，SKIPIF1<0，則異面直線SKIPIF1<0與SKIPIF1<0所成角的余弦值為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0例題4．（2022·云南·麗江市教育科學(xué)研究所高二期末）如圖，SKIPIF1<0是正方體的一個(gè)“直角尖”SKIPIF1<0（SKIPIF1<0兩兩垂直且相等）棱SKIPIF1<0的中點(diǎn)，SKIPIF1<0是SKIPIF1<0中點(diǎn)，SKIPIF1<0是SKIPIF1<0上的一個(gè)動(dòng)點(diǎn)，連接SKIPIF1<0，則當(dāng)SKIPIF1<0與SKIPIF1<0所成角為最小時(shí)，SKIPIF1<0_________．角度2：用向量法求異面直線所成角典型例題例題1．（2022·山東德州·高一期末）已知SKIPIF1<0?SKIPIF1<0?SKIPIF1<0?SKIPIF1<0分別是正方體SKIPIF1<0，邊SKIPIF1<0，SKIPIF1<0，SKIPIF1<0，SKIPIF1<0的中點(diǎn)，則異面直線SKIPIF1<0與SKIPIF1<0所成角的余弦值為_(kāi)__________.例題2．（2022·河南省蘭考縣第一高級(jí)中學(xué)模擬預(yù)測(cè)（理））已知三棱柱SKIPIF1<0的底面是邊長(zhǎng)為2的等邊三角形，側(cè)棱長(zhǎng)為2，SKIPIF1<0為SKIPIF1<0的中點(diǎn)，若SKIPIF1<0，則異面直線SKIPIF1<0與SKIPIF1<0所成角的余弦值為_(kāi)_____．角度3：用向量法解決線面角的問(wèn)題（定值+探索性問(wèn)題（最值，求參數(shù)））典型例題例題1．（2022·全國(guó)·高二單元測(cè)試）如圖，四棱錐SKIPIF1<0中，底面SKIPIF1<0為平行四邊形，且SKIPIF1<0，SKIPIF1<0，SKIPIF1<0，若二面角SKIPIF1<0為SKIPIF1<0，則SKIPIF1<0與平面SKIPIF1<0所成角的正弦值為_(kāi)_________．例題2．（2022·黑龍江·大慶實(shí)驗(yàn)中學(xué)高一期末）如圖，在四棱錐SKIPIF1<0中，底面SKIPIF1<0為菱形，SKIPIF1<0，SKIPIF1<0為SKIPIF1<0的中點(diǎn)，SKIPIF1<0．(1)點(diǎn)SKIPIF1<0在線段SKIPIF1<0上，SKIPIF1<0，求證：SKIPIF1<0平面SKIPIF1<0；(2)在（1）的條件下，若SKIPIF1<0，求直線SKIPIF1<0和平面SKIPIF1<0所成角的余弦值．例題3．（2022·天津一中高一期末）如圖，SKIPIF1<0且SKIPIF1<0，SKIPIF1<0，SKIPIF1<0且SKIPIF1<0，SKIPIF1<0且SKIPIF1<0．SKIPIF1<0平面SKIPIF1<0，SKIPIF1<0．(1)若SKIPIF1<0為SKIPIF1<0的中點(diǎn)，SKIPIF1<0為SKIPIF1<0的中點(diǎn)，求證：SKIPIF1<0平面SKIPIF1<0；(2)求平面SKIPIF1<0與平面SKIPIF1<0的夾角的正弦值；(3)若點(diǎn)SKIPIF1<0在線段SKIPIF1<0上，且直線SKIPIF1<0與平面SKIPIF1<0所成的角為SKIPIF1<0，求線段SKIPIF1<0的長(zhǎng)．例題4．（2022·湖北·鄂州市教學(xué)研究室高二期末）蓮花山位于鄂州市洋瀾湖畔．蓮花山，山連九峰，狀若金色蓮初開(kāi)，獨(dú)展靈秀，故而得名．這里三面環(huán)湖，通匯長(zhǎng)江，山巒疊翠，煙波浩渺．旅游區(qū)管委會(huì)計(jì)劃在山上建設(shè)別致涼亭供游客歇腳，如圖①為該涼亭的實(shí)景效果圖，圖②為設(shè)計(jì)圖，該涼亭的支撐柱高為3SKIPIF1<0m，頂部為底面邊長(zhǎng)為2的正六棱錐，且側(cè)面與底面所成的角都是SKIPIF1<0．(1)求該涼亭及其內(nèi)部所占空間的大?。?2)在直線SKIPIF1<0上是否存在點(diǎn)SKIPIF1<0，使得直線SKIPIF1<0與平面SKIPIF1<0所成角的正弦值為SKIPIF1<0？若存在，請(qǐng)確定點(diǎn)SKIPIF1<0的位置；若不存在，請(qǐng)說(shuō)明理由．角度4：用向量法解決二面角的問(wèn)題（定值+探索性問(wèn)題（最值，求參數(shù)））典型例題例題1．（2022·吉林·長(zhǎng)春市實(shí)驗(yàn)中學(xué)高一期末）如圖在三棱錐SKIPIF1<0中，SKIPIF1<0，SKIPIF1<0且SKIPIF1<0．(1)求證：平面SKIPIF1<0平面SKIPIF1<0(2)若SKIPIF1<0為SKIPIF1<0中點(diǎn)，求平面SKIPIF1<0與平面SKIPIF1<0所成銳二面角的余弦值．例題2．（2022·四川雅安·高二期末（理））如圖（一）四邊形SKIPIF1<0是等腰梯形，SKIPIF1<0，SKIPIF1<0，SKIPIF1<0，SKIPIF1<0，過(guò)SKIPIF1<0點(diǎn)作SKIPIF1<0，垂足為SKIPIF1<0點(diǎn)，將SKIPIF1<0沿SKIPIF1<0折到SKIPIF1<0位置如圖（二），且SKIPIF1<0．(1)證明：平面SKIPIF1<0平面EBCD；(2)已知點(diǎn)SKIPIF1<0在棱SKIPIF1<0上，且SKIPIF1<0，求二面角SKIPIF1<0的余弦值．例題3．（2022·全國(guó)·高三專(zhuān)題練習(xí)）四棱雉SKIPIF1<0中，SKIPIF1<0平面SKIPIF1<0，底面SKIPIF1<0是等腰梯形，且SKIPIF1<0，點(diǎn)SKIPIF1<0在棱SKIPIF1<0上.(1)當(dāng)SKIPIF1<0是棱SKIPIF1<0的中點(diǎn)時(shí)，求證:SKIPIF1<0平面SKIPIF1<0;(2)當(dāng)直線SKIPIF1<0與平面SKIPIF1<0所成角SKIPIF1<0最大時(shí)，求二面角SKIPIF1<0的大小.例題4．（2022·江蘇徐州·高二期末）如圖，已知SKIPIF1<0垂直于梯形SKIPIF1<0所在的平面，矩形SKIPIF1<0的對(duì)角線交于點(diǎn)SKIPIF1<0，SKIPIF1<0為SKIPIF1<0的中點(diǎn)，SKIPIF1<0，SKIPIF1<0.(1)求證：SKIPIF1<0SKIPIF1<0平面SKIPIF1<0；(2)求二面角SKIPIF1<0的余弦值；(3)在線段SKIPIF1<0上是否存在一點(diǎn)SKIPIF1<0，使得SKIPIF1<0與平面SKIPIF1<0所成角的大小為SKIPIF1<0？若存在，求出SKIPIF1<0的長(zhǎng)；若不存在，說(shuō)明理由.題型三：距離問(wèn)題角度1：點(diǎn)SKIPIF1<0到直線SKIPIF1<0的距離典型例題例題1．（2022·湖南益陽(yáng)·高二期末）在棱長(zhǎng)為1的正方體SKIPIF1<0中，SKIPIF1<0為SKIPIF1<0的中點(diǎn)，則點(diǎn)SKIPIF1<0到直線SKIPIF1<0的距離為（

）A．SKIPIF1<0 B．1 C．SKIPIF1<0 D．SKIPIF1<0例題2．（2022·北京·二模）如圖，已知正方體SKIPIF1<0的棱長(zhǎng)為1，則線段SKIPIF1<0上的動(dòng)點(diǎn)SKIPIF1<0到直線SKIPIF1<0的距離的最小值為（

）A．1 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0角度2：點(diǎn)SKIPIF1<0到平面SKIPIF1<0的距離（等體積法）典型例題例題1．（2022·四川廣安·模擬預(yù)測(cè)（文））如圖，四棱錐SKIPIF1<0中，底面SKIPIF1<0為直角梯形，其中SKIPIF1<0，SKIPIF1<0，面SKIPIF1<0面SKIPIF1<0，且SKIPIF1<0，點(diǎn)SKIPIF1<0在棱SKIPIF1<0上．(1)若SKIPIF1<0，求證：SKIPIF1<0平面SKIPIF1<0．(2)當(dāng)SKIPIF1<0平面SKIPIF1<0時(shí)，求點(diǎn)SKIPIF1<0到平面SKIPIF1<0的距離．例題2．（2022·云南保山·高一期末）如圖，在四棱錐SKIPIF1<0，四邊形SKIPIF1<0正方形，SKIPIF1<0平面SKIPIF1<0．SKIPIF1<0，SKIPIF1<0，點(diǎn)SKIPIF1<0是SKIPIF1<0的中點(diǎn)．(1)求證：SKIPIF1<0平面SKIPIF1<0；(2)求點(diǎn)SKIPIF1<0到平面SKIPIF1<0的距離．角度3：點(diǎn)SKIPIF1<0到平面SKIPIF1<0的距離（向量法）典型例題例題1．（2022·江蘇·淮安市淮安區(qū)教師發(fā)展中心學(xué)科研訓(xùn)處高二期中）將邊長(zhǎng)為SKIPIF1<0的正方形SKIPIF1<0沿對(duì)角線SKIPIF1<0折成直二面角，則點(diǎn)SKIPIF1<0到平面SKIPIF1<0的距離為_(kāi)_____．例題2．（2022·江蘇·南京市第一中學(xué)高二階段練習(xí)）如圖，四棱錐SKIPIF1<0的底面是正方形，SKIPIF1<0底面SKIPIF1<0，SKIPIF1<0為SKIPIF1<0的中點(diǎn)，若SKIPIF1<0，則點(diǎn)SKIPIF1<0到平面SKIPIF1<0的距離為_(kāi)__________.例題3．（2022·全國(guó)·高二單元測(cè)試）在如圖所示的幾何體中，四邊形SKIPIF1<0為矩形，SKIPIF1<0平面SKIPIF1<0，SKIPIF1<0，點(diǎn)SKIPIF1<0為棱SKIPIF1<0的中點(diǎn)．(1)求證：SKIPIF1<0平面SKIPIF1<0；(2)求直線SKIPIF1<0與平面SKIPIF1<0所成角的正弦值；(3)求點(diǎn)SKIPIF1<0到平面SKIPIF1<