第七章立體幾何與空間向量[知識網絡][命題方向]1．空間向量與立體幾何是高考的熱點，難度以中檔或偏難為主，題型涵蓋選擇，填空和解答題，其中解答題必考，總分值大約22分．(1)從近幾年高考情況來看，本章內容在客觀題中主要考查空間幾何體的表面積、體積以及與球相關的問題．(2)在解答題中主要考查線、面位置關系的證明及空間角的計算，如2023年新高考Ⅰ卷第18題第(1)問考查了證明線線平行的方法，第(2)問考查了面面角，建系是解題的關鍵．可以用試點法解決，坐標原點可以是頂點、底面邊上的一點或垂心、重心，要根據空間幾何體的結構特征確定，并且建系后，要能求出有關點的坐標，難度不大，但需要計算細心，防止得分不全的情況．(3)本章重點考查的核心素養為邏輯推理、直觀想象和數學運算．2．空間向量與立體幾何考查的主要內容有：(1)以空間幾何體或與球的切、接為背景考查幾何體的結構特征、表面積、體積、球的性質等；(2)以空間幾何體為載體考查空間線、面位置關系的證明以及空間角和距離的計算；(3)以空間向量為輔助工具解決空間角的計算與線、面位置關系的證明．探究1(人教A版必修第二冊P102)圓柱可以由矩形旋轉得到，圓錐可以由直角三角形旋轉得到．圓臺是否也可以由平面圖形旋轉得到？如果可以，由什么平面圖形旋轉得到？如何旋轉？_________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________探究2(人教A版必修第二冊P117)圓柱、圓錐、圓臺的體積公式之間有什么關系？結合棱柱、棱錐、棱臺的體積公式，你能將它們統一成柱體、錐體、臺體的體積公式嗎？柱體、錐體、臺體的體積公式之間又有怎樣的關系？_________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________(2022·新高考Ⅰ卷)南水北調工程緩解了北方一些地區水資源短缺問題，其中一部分水蓄入某水庫．已知該水庫水位為海拔148.5m時，相應水面的面積為140.0km2；水位為海拔157.5m時，相應水面的面積為180.0km2.將該水庫在這兩個水位間的形狀看作一個棱臺，則該水庫水位從海拔148.5m上升到157.5m時，增加的水量約為(eq\r(7)≈2.65)()A．1.0×109m3 B．1.2×109m3C．1.4×109m3 D．1.6×109m3點評本題與教材思考題都考查臺體的體積公式．探究3(人教A版必修第二冊P151)兩條相交直線可以確定一個平面，兩條平行直線也可以確定一個平面，那么定理“如果一條直線與一個平面內的兩條相交直線垂直，那么該直線與此平面垂直”中的“兩條相交直線”可以改為“兩條平行直線”嗎？你能從向量的角度解釋原因嗎？如果改為“無數條直線”呢？____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________探究4(人教A版必修第二冊P160)設平面α⊥平面β，點P在平面α內，過點P作平面β的垂線a，直線a與平面α具有什么位置關系？____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________典題1(人教A版必修第二冊P116練習T4)求證：直三棱柱的任意兩個側面的面積和大于第三個側面的面積．______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________典題2(人教A版必修第二冊P120習題8.3T3)如圖，一個三棱柱形容器中盛有水，側棱AA1＝8.若側面AA1B1B水平放置時，水面恰好過AC，BC，A1C1，B1C1的中點．那么當底面ABC水平放置時，水面高為多少？_________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________(2021·北京卷)對24小時內降水在平地上的積水厚度(mm)進行如下定義：0～1010～2525～5050～100小雨中雨大雨暴雨小明用一個圓錐形容器接了24小時的雨水，則這一天的雨水屬于哪個等級()A．小雨 B．中雨C．大雨 D．暴雨點評本題與教材習題的解題思路完全一樣，都是利用變換兩類幾何體的體積相等求水面的高度．典題3(人教A版必修第二冊P120習題8.3T4)如圖，圓錐PO的底面直徑和高均是a，過PO的中點O′作平行于底面的截面，以該截面為底面挖去一個圓柱，求剩下幾何體的表面積和體積．______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________(2022·全國甲卷)甲、乙兩個圓錐的母線長相等，側面展開圖的圓心角之和為2π，側面積分別為S甲和S乙，體積分別為V甲和V乙．若eq\f(S甲,S乙)＝2，則eq\f(V甲,V乙)＝()A.eq\r(5) B．2eq\r(2)C.eq\r(10) D.eq\f(5\r(10),4)點評本題與教材習題都考查了空間幾何體體積的求解．典題4(人教A版必修第二冊P120習題8.3T9)如圖是一個獎杯的三視圖，試根據獎杯的三視圖計算它的表面積和體積．(可用計算工具，尺寸如圖，單位：cm，π取3.14，結果取整數．)______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________點評千萬不要認為三視圖不在高考的范圍內，此內容在初中學過，在教材中也出現了，高考試題中出現了正常．典題5(人教A版必修第二冊P137例2)求證：空間四邊形相鄰兩邊中點的連線平行于經過另外兩邊的平面．_________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________典題6(人教A版必修第二冊P138例3)如圖所示的一塊木料中，棱BC平行于面A′C′.(1)要經過面A′C′內的一點P和棱BC將木料鋸開，在木料表面應該怎樣畫線？(2)所畫的線與平面AC是什么位置關系？______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________典題7(人教A版必修第二冊P144習題8.5T8)如圖，直線AA′，BB′，CC′相交于點O，AO＝A′O，BO＝B′O，CO＝C′O，求證：平面ABC∥平面A′B′C′.___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________典題8(人教A版必修第二冊P144習題8.5T9)如圖，E，E′分別為長方體ABCD－A′B′C′D′的棱AD，A′D′的中點，求證：∠BEC＝∠B′E′C′.___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________典題9(人教A版必修第二冊P145習題8.5T13)如圖，α∥β∥γ，直線a與b分別交α，β，γ于點A，B，C和點D，E，F，求證：eq\f(AB,BC)＝eq\f(DE,EF).__________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________典題10(人教A版必修第二冊P145習題8.5T15)如圖，透明塑料制成的長方體容器ABCD－A1B1C1D1內灌進一些水，固定容器底面一邊BC于地面上，再將容器傾斜．隨著傾斜度的不同，有下面五個命題：(1)有水的部分始終呈棱柱形；(2)沒有水的部分始終呈棱柱形；(3)水面EFGH所在四邊形的面積為定值；(4)棱A1D1始終與水面所在平面平行；(5)當容器傾斜如圖(3)所示時，BE·BF是定值．其中所有正確命題的序號是________，為什么？_______________________________________________________________________________________________________________________________________________________________________________________________________________典題11(人教A版必修第二冊P147例1)如圖，已知正方體ABCD－A′B′C′D′.(1)哪些棱所在的直線與直線AA′垂直？(2)求直線BA′與CC′所成的角的大小．(3)求直線BA′與AC所成的角的大小．____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________(2021·全國乙卷)在正方體ABCD－A1B1C1D1中，P為B1D1的中點，則直線PB與AD1所成的角為()A.eq\f(π,2) B.eq\f(π,3)C.eq\f(π,4) D.eq\f(π,6)點評本題和教材例題類似，無論是問題的載體，還是考查角度、解題方法完全一樣．其實就是在教材例題的基礎上稍微換了一條直線而已．典題12(人教A版必修第二冊P147例2)如圖，在正方體ABCD－A1B1C1D1中，O1為底面A1B1C1D1的中心．求證：AO1⊥BD.____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________(多選)(2021·新高考Ⅱ卷)如圖，在正方體中，O為底面的中心，P為所在棱的中點，M，N為正方體的頂點，則滿足MN⊥OP的是()點評本題和教材例題類似，考查角度都是證明線線垂直，其方法都是通過線面垂直或利用空間向量解決問題．典題13(人教A版必修第二冊P151例3)求證：如果兩條平行直線中的一條直線垂直于一個平面，那么另一條直線也垂直于這個平面．已知：如圖，a∥b，a⊥α，求證：b⊥α.________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________典題14(人教A版必修第二冊P152練習T4)過△ABC所在平面α外一點P作PO⊥α.垂足為O.連接PA，PB，PC.(1)若PA＝PB＝PC，則點O是△ABC的________心．(2)若PA＝PB＝PC，∠C＝90°，則點O是AB邊的________點．(3)若PA⊥PB，PB⊥PC，PC⊥PA，垂足都為P，則點O是△ABC的________心．_______________________________________________________________________________________________________________________________________________________________________________________________________________典題15(人教A版必修第二冊P154例6)推導棱臺的體積公式V棱臺＝eq\f(1,3)h(S′＋eq\r(S′S)＋S)．其中S′，S分別是棱臺的上、下底面面積，h是高．___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________(2021·新高考Ⅱ卷)正四棱臺的上、下底面的邊長分別為2，4，側棱長為2，則其體積為()A．20＋12eq\r(3) B．28eq\r(2)C.eq\f(56,3) D.eq\f(28\r(2),3)點評本題是教材例題所推導的公式的具體應用，可由四棱臺的幾何特征算出該幾何體的高及上下底面面積，再由棱臺的體積公式即可得解．典題16(人教A版必修第二冊P158例8)如圖，AB是⊙O的直徑，PA垂直于⊙O所在的平面，C是圓周上不同于A，B的任意一點，求證：平面PAC⊥平面PBC.________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________典題17(人教A版必修第二冊P164習題8.6T19)如圖，在直三棱柱ABC－A1B1C1中，∠ABC＝90°，AA1＝AB，求證：A1C⊥AB1._____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________典題18(人教A版必修第二冊P165習題8.6T20)如圖，AB是⊙O的直徑，點C是⊙O上的動點，過動點C的直線VC垂直于⊙O所在平面，D，E分別是VA，VC的中點．判斷直線DE與平面VBC的位置關系，并說明理由．__________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________典題19(人教A版必修第二冊P165習題8.6T21)如圖，在四棱錐P－ABCD中，底面ABCD為正方形，PA⊥底面ABCD，PA＝AB，E為線段PB的中點，F為線段BC上的動點．平面AEF與平面PBC是否互相垂直？如果垂直，請證明；如果不垂直，請說明理由．__________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________典題20(人教A版必修第二冊P170復習參考題8T10)如圖，在邊長為2的正方形ABCD中，點E是AB的中點，點F是BC的中點，將△AED，△BEF，△DCF分別沿DE，EF，DF折起，使A，B，C三點重合于點A′.(1)求證：A′D⊥EF；(2)求三棱錐A′－EFD的體積．___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________典題21(人教A版必修第二冊P171復習參考題8T13)如圖，在三棱錐P－ABC中，∠ACB＝90°，PA⊥底面ABC.(1)求證：平面PAC⊥平面PBC；(2)若AC＝BC＝PA，M是PB的中點，求AM與平面PBC所成角的正切值．__________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________典題22(人教A版必修第二冊P171復習參考題8T14)如圖，在四棱錐P－ABCD中，底面ABCD為正方形，側面PAD是正三角形，側面PAD⊥底面ABCD，M是PD的中點．(1)求證：AM⊥平面PCD；(2)求側面PBC與底面ABCD所成二面角的余弦值．_____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________典題23(人教A版選擇性必修第一冊P34例6)如圖，在棱長為1的正方體ABCD－A1B1C1D1中，E為線段A1B1的中點，F為線段AB的中點．(1)求點B到直線AC1的距離；(2)求直線FC到平面AEC1的距離．_________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________