新高考數(shù)學(xué)二輪復(fù)習(xí)分層訓(xùn)練專(zhuān)題18 空間向量在立體幾何中的應(yīng)用（角和距離） （原卷版）

答案第=page11頁(yè)，共=sectionpages22頁(yè)專(zhuān)題18空間向量在立體幾何中的應(yīng)用（角和距離）【練基礎(chǔ)】一、單選題1．（2022·全國(guó)·統(tǒng)考高考真題）在正方體SKIPIF1<0中，E，F(xiàn)分別為SKIPIF1<0的中點(diǎn)，則（

）A．平面SKIPIF1<0平面SKIPIF1<0 B．平面SKIPIF1<0平面SKIPIF1<0C．平面SKIPIF1<0平面SKIPIF1<0 D．平面SKIPIF1<0平面SKIPIF1<02．（2023·全國(guó)·高三專(zhuān)題練習(xí)）下圖為正三棱柱SKIPIF1<0的一個(gè)展開(kāi)圖，若A，SKIPIF1<0，SKIPIF1<0，D，SKIPIF1<0，SKIPIF1<0六點(diǎn)在同一個(gè)圓周上，則在原正三棱柱中，直線AE和直線BF所成角的余弦值是（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<03．（2022秋·湖北黃岡·高三校考期中）平行六面體SKIPIF1<0中，SKIPIF1<0，則SKIPIF1<0與底面SKIPIF1<0所成的線面角的正弦值是（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<04．（2023·全國(guó)·高三專(zhuān)題練習(xí)）在矩形ABCD中，O為BD中點(diǎn)且SKIPIF1<0，將平面ABD沿對(duì)角線BD翻折至二面角SKIPIF1<0為90°，則直線AO與CD所成角余弦值為（

）A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<05．（2023·全國(guó)·高三專(zhuān)題練習(xí)）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)P到直線SKIPIF1<0的距離的最小值為（

）A．1 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<06．（2023·全國(guó)·高三專(zhuān)題練習(xí)）正方體SKIPIF1<0棱長(zhǎng)為SKIPIF1<0，動(dòng)點(diǎn)SKIPIF1<0在線段SKIPIF1<0上（含端點(diǎn)），以下結(jié)論不正確的為（

）A．三棱錐SKIPIF1<0的體積為定值SKIPIF1<0B．過(guò)SKIPIF1<0，SKIPIF1<0，SKIPIF1<0三點(diǎn)若可作正方體的截面，則截面圖形為三角形或平面四邊形C．當(dāng)點(diǎn)SKIPIF1<0和SKIPIF1<0重合時(shí)，三棱錐SKIPIF1<0的外接球體積為SKIPIF1<0D．直線SKIPIF1<0與面SKIPIF1<0所成角的正弦值的范圍為SKIPIF1<07．（2022·河南省直轄縣級(jí)單位·統(tǒng)考二模）如圖，已知AB是圓柱底面圓的一條直徑，OP是圓柱的一條母線，C為底面圓上一點(diǎn)，且SKIPIF1<0，SKIPIF1<0，則直線PC與平面PAB所成角的正弦值為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<08．（2023·全國(guó)·高三專(zhuān)題練習(xí)）在中國(guó)古代數(shù)學(xué)著作《九章算術(shù)》中記載了一種稱為“曲池”的幾何體，該幾何體的上?下底面平行，且均為扇環(huán)形（扇環(huán)是指圓環(huán)被扇形截得的部分）.現(xiàn)有一個(gè)如圖所示的曲池，它的高為2，SKIPIF1<0，SKIPIF1<0，SKIPIF1<0，SKIPIF1<0均與曲池的底面垂直，底面扇環(huán)對(duì)應(yīng)的兩個(gè)圓的半徑分別為1和2，對(duì)應(yīng)的圓心角為90°，則圖中異面直線SKIPIF1<0與SKIPIF1<0所成角的余弦值為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0二、多選題9．（2023·云南·統(tǒng)考模擬預(yù)測(cè)）如圖，在正方體SKIPIF1<0中，E?F?G分別為SKIPIF1<0的中點(diǎn)，則（

）A．SKIPIF1<0 B．SKIPIF1<0與SKIPIF1<0所成角為SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<0平面SKIPIF1<010．（2023·全國(guó)·模擬預(yù)測(cè)）已知四棱錐SKIPIF1<0的頂點(diǎn)都在一個(gè)表面積為SKIPIF1<0的球面上，SKIPIF1<0平面ABCD，底面ABCD是正方形，SKIPIF1<0，則（

）A．SKIPIF1<0B．SKIPIF1<0C．直線PC與直線AB所成角的大小為SKIPIF1<0D．直線PC與平面PAB所成角的大小為SKIPIF1<011．（2023·山東威?！そy(tǒng)考一模）在棱長(zhǎng)為1的正方體SKIPIF1<0中，點(diǎn)P滿足SKIPIF1<0，SKIPIF1<0，SKIPIF1<0，則(

)A．當(dāng)SKIPIF1<0時(shí)，SKIPIF1<0的最小值為SKIPIF1<0B．當(dāng)SKIPIF1<0時(shí)，有且僅有一點(diǎn)P滿足SKIPIF1<0C．當(dāng)SKIPIF1<0時(shí)，有且僅有一點(diǎn)P滿足到直線SKIPIF1<0的距離與到平面ABCD的距離相等D．當(dāng)SKIPIF1<0時(shí)，直線AP與SKIPIF1<0所成角的大小為定值12．（2023·廣東梅州·統(tǒng)考一模）如圖，在直三棱柱SKIPIF1<0中，SKIPIF1<0，SKIPIF1<0，SKIPIF1<0，SKIPIF1<0為棱SKIPIF1<0的中點(diǎn)；SKIPIF1<0為棱SKIPIF1<0上的動(dòng)點(diǎn)（含端點(diǎn)），過(guò)點(diǎn)A?SKIPIF1<0?SKIPIF1<0作三棱柱的截面SKIPIF1<0，且SKIPIF1<0交SKIPIF1<0于SKIPIF1<0，則（

）A．線段SKIPIF1<0的最小值為SKIPIF1<0 B．棱SKIPIF1<0上的不存在點(diǎn)SKIPIF1<0，使得SKIPIF1<0平面SKIPIF1<0C．棱SKIPIF1<0上的存在點(diǎn)SKIPIF1<0，使得SKIPIF1<0 D．當(dāng)SKIPIF1<0為棱SKIPIF1<0的中點(diǎn)時(shí)，SKIPIF1<0三、填空題13．（2023·廣東江門(mén)·統(tǒng)考一模）已知直線l過(guò)點(diǎn)SKIPIF1<0，且直線l的一個(gè)方向向量為SKIPIF1<0，則坐標(biāo)原點(diǎn)O到直線l的距離d為_(kāi)__________.14．（2023·上?！そy(tǒng)考模擬預(yù)測(cè)）正方體SKIPIF1<0的邊長(zhǎng)為1，點(diǎn)SKIPIF1<0分別為SKIPIF1<0邊的中點(diǎn)，SKIPIF1<0是側(cè)面SKIPIF1<0上動(dòng)點(diǎn)，若直線SKIPIF1<0與面SKIPIF1<0的交點(diǎn)位于SKIPIF1<0內(nèi)（包括邊界），則所有滿足要求的點(diǎn)SKIPIF1<0構(gòu)成的圖形面積為_(kāi)_________.15．（2022·湖南永州·統(tǒng)考一模）在四棱錐SKIPIF1<0中，平面SKIPIF1<0平面SKIPIF1<0，四邊形SKIPIF1<0為等腰梯形，SKIPIF1<0為等邊三角形，SKIPIF1<0，則四棱錐SKIPIF1<0的外接球球心SKIPIF1<0到平面SKIPIF1<0的距離是___________.16．（2023春·北京·高三北京市陳經(jīng)綸中學(xué)?？奸_(kāi)學(xué)考試）如圖，在棱長(zhǎng)為2的正方體ABCD﹣A1B1C1D1中，點(diǎn)E為棱CD的中點(diǎn)，點(diǎn)F為底面ABCD內(nèi)一點(diǎn)，給出下列三個(gè)論斷：①A1F⊥BE；②A1F＝3；③S△ADF＝2S△ABF.以其中的一個(gè)論斷作為條件，另一個(gè)論斷作為結(jié)論，寫(xiě)出一個(gè)正確的命題：__.四、解答題17．（2023·全國(guó)·高三專(zhuān)題練習(xí)）如圖，在四棱錐P-ABCD中，SKIPIF1<0，且SKIPIF1<0，底面ABCD是邊長(zhǎng)為2的菱形，SKIPIF1<0．(1)證明：平面PAC⊥平面ABCD；(2)若SKIPIF1<0，求平面PAB與平面PBC夾角的余弦值．18．（2023·山東泰安·統(tǒng)考一模）在如圖所示的幾何體中，底面ABCD是邊長(zhǎng)為6的正方形，SKIPIF1<0，SKIPIF1<0，SKIPIF1<0，SKIPIF1<0，點(diǎn)P，Q分別在棱GD，BC上，且SKIPIF1<0，SKIPIF1<0，SKIPIF1<0.(1)證明：SKIPIF1<0平面ABCD；(2)設(shè)H為線段GC上一點(diǎn)，且三棱錐SKIPIF1<0的體積為18，求平面ACH與平面ADH夾角的余弦值.【提能力】一、單選題19．（2023·全國(guó)·高三專(zhuān)題練習(xí)）如圖，在三棱錐SKIPIF1<0中，SKIPIF1<0平面ABC，SKIPIF1<0是邊長(zhǎng)為2的正三角形，SKIPIF1<0，E，F(xiàn)分別為MA，MC的中點(diǎn)，則異面直線BE與AF所成角的余弦值為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<020．（2022·陜西咸陽(yáng)·武功縣普集高級(jí)中學(xué)?？家荒＃┤鐖D，在直二面角SKIPIF1<0中，SKIPIF1<0是直線SKIPIF1<0上兩點(diǎn)，點(diǎn)SKIPIF1<0，點(diǎn)SKIPIF1<0，且SKIPIF1<0，SKIPIF1<0，SKIPIF1<0，那么直線SKIPIF1<0與直線SKIPIF1<0所成角的余弦值為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<021．（2023·湖南衡陽(yáng)·?？寄M預(yù)測(cè)）如圖所示，正方體SKIPIF1<0中，點(diǎn)SKIPIF1<0為底面SKIPIF1<0的中心，點(diǎn)SKIPIF1<0在側(cè)面SKIPIF1<0的邊界及其內(nèi)部移動(dòng)，若SKIPIF1<0，則異面直線SKIPIF1<0與SKIPIF1<0所成角的余弦值的最大值為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<022．（2023·全國(guó)·高三專(zhuān)題）如圖正方體SKIPIF1<0，中，點(diǎn)SKIPIF1<0、SKIPIF1<0分別是SKIPIF1<0、SKIPIF1<0的中點(diǎn)，SKIPIF1<0為正方形SKIPIF1<0的中心，則（

）A．直線SKIPIF1<0與SKIPIF1<0是異面直線 B．直線SKIPIF1<0與SKIPIF1<0是相交直線C．直線SKIPIF1<0與SKIPIF1<0互相垂直 D．直線SKIPIF1<0與SKIPIF1<0所成角的余弦值為SKIPIF1<023．（2021·山西呂梁·統(tǒng)考一模）如圖正三棱柱SKIPIF1<0的各棱長(zhǎng)相等，SKIPIF1<0為SKIPIF1<0的中點(diǎn)，則異面直線SKIPIF1<0與SKIPIF1<0所成的角為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<024．（2022·浙江·高三專(zhuān)題）已知正方體SKIPIF1<0的棱長(zhǎng)為2，點(diǎn)E，F(xiàn)在平面SKIPIF1<0內(nèi)，若SKIPIF1<0，SKIPIF1<0，則下列選項(xiàng)中錯(cuò)誤的是（

）A．點(diǎn)E的軌跡是圓的一部分 B．點(diǎn)F的軌跡是一條線段C．SKIPIF1<0的最小值為SKIPIF1<0 D．SKIPIF1<0與平面SKIPIF1<0所成角的正弦值的最大值為SKIPIF1<025．（2023·全國(guó)·高三專(zhuān)題練習(xí)）如圖，四邊形SKIPIF1<0中，SKIPIF1<0．現(xiàn)將SKIPIF1<0沿SKIPIF1<0折起，當(dāng)二面角SKIPIF1<0處于SKIPIF1<0過(guò)程中，直線SKIPIF1<0與SKIPIF1<0所成角的余弦值取值范圍是（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<026．（2022·全國(guó)·高三專(zhuān)題練習(xí)）已知長(zhǎng)方體SKIPIF1<0中，SKIPIF1<0，點(diǎn)SKIPIF1<0在線段SKIPIF1<0上，SKIPIF1<0平面SKIPIF1<0過(guò)線段SKIPIF1<0的中點(diǎn)以及點(diǎn)SKIPIF1<0、SKIPIF1<0，現(xiàn)有如下說(shuō)法：（1）SKIPIF1<0，使得SKIPIF1<0；（2）若SKIPIF1<0，則平面SKIPIF1<0截長(zhǎng)方體SKIPIF1<0所得截面為平行四邊形；（3）若SKIPIF1<0，SKIPIF1<0，則平面SKIPIF1<0截長(zhǎng)方體SKIPIF1<0所得截面的面積為SKIPIF1<0以上說(shuō)法正確的個(gè)數(shù)為（

）A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0二、多選題27．（2022·全國(guó)·模擬預(yù)測(cè)）已知正方體SKIPIF1<0中，SKIPIF1<0，SKIPIF1<0，SKIPIF1<0分別為棱AB，BC的中點(diǎn)，過(guò)點(diǎn)E，F(xiàn)作正方體的截面，則下列說(shuō)法正確的是（

）A．若截面過(guò)點(diǎn)SKIPIF1<0，則截面周長(zhǎng)為SKIPIF1<0B．若點(diǎn)SKIPIF1<0是線段SKIPIF1<0上的動(dòng)點(diǎn)（不含端點(diǎn)），則SKIPIF1<0的最小值為SKIPIF1<0C．若截面是正六邊形，則直線SKIPIF1<0與截面垂直D．若截面是正六邊形，S，T是截面上兩個(gè)不同的動(dòng)點(diǎn)，設(shè)直線SKIPIF1<0與直線ST所成角的最小值為SKIPIF1<0，則SKIPIF1<028．（2022秋·吉林長(zhǎng)春·高三長(zhǎng)春市第二實(shí)驗(yàn)中學(xué)校考期末）如圖，在平行四邊形SKIPIF1<0中，SKIPIF1<0，SKIPIF1<0分別為SKIPIF1<0的中點(diǎn)，沿SKIPIF1<0將SKIPIF1<0折起到SKIPIF1<0的位置（SKIPIF1<0不在平面SKIPIF1<0上），在折起過(guò)程中，下列說(shuō)法不正確的是（

）A．若SKIPIF1<0是SKIPIF1<0的中點(diǎn)，則SKIPIF1<0平面SKIPIF1<0B．存在某位置，使SKIPIF1<0C．當(dāng)二面角SKIPIF1<0為直二面角時(shí)，三棱錐SKIPIF1<0外接球的表面積為SKIPIF1<0D．直線SKIPIF1<0和平面SKIPIF1<0所成的角的最大值為SKIPIF1<029．（2023·云南曲靖·曲靖一中校考模擬預(yù)測(cè)）如圖，在棱長(zhǎng)為SKIPIF1<0的正方體中，下列結(jié)論成立的是（

）A．若點(diǎn)SKIPIF1<0是平面SKIPIF1<0的中心，則點(diǎn)SKIPIF1<0到直線SKIPIF1<0的距離為SKIPIF1<0B．二面角SKIPIF1<0的正切值為SKIPIF1<0C．直線SKIPIF1<0與平面SKIPIF1<0所成的角為SKIPIF1<0D．若SKIPIF1<0是平面SKIPIF1<0的中心，點(diǎn)SKIPIF1<0是平面SKIPIF1<0的中心，則SKIPIF1<0面SKIPIF1<030．（2023·全國(guó)·高三專(zhuān)題練習(xí)）在直三棱柱SKIPIF1<0中，SKIPIF1<0，SKIPIF1<0，SKIPIF1<0為SKIPIF1<0的中點(diǎn)，點(diǎn)SKIPIF1<0是線段SKIPIF1<0上的點(diǎn)，則下列說(shuō)法正確的是（

）A．SKIPIF1<0B．存在點(diǎn)SKIPIF1<0，使得直線SKIPIF1<0與SKIPIF1<0所成的角是SKIPIF1<0C．當(dāng)點(diǎn)SKIPIF1<0是線段SKIPIF1<0的中點(diǎn)時(shí)，三棱錐SKIPIF1<0外接球的表面積是SKIPIF1<0D．當(dāng)點(diǎn)SKIPIF1<0是線段SKIPIF1<0的中點(diǎn)時(shí)，直線SKIPIF1<0與平面SKIPIF1<0所成角的正切值為SKIPIF1<0．三、填空題31．（2022·全國(guó)·高三專(zhuān)題練習(xí)）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，D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)_____．32．（2023·全國(guó)·高三專(zhuān)題練習(xí)）在棱長(zhǎng)為1的正方體SKIPIF1<0中，點(diǎn)P是對(duì)角線SKIPIF1<0的動(dòng)點(diǎn)（點(diǎn)P與SKIPIF1<0不重合），則下列結(jié)論正確的有___________.①存在點(diǎn)P，使得平面SKIPIF1<0SKIPIF1<0平面SKIPIF1<0；②存在點(diǎn)P，使得SKIPIF1<0平面SKIPIF1<0；③SKIPIF1<0分別是SKIPIF1<0在平面SKIPIF1<0，平面SKIPIF1<0上的正投影圖形的面積，對(duì)任意的點(diǎn)P都有SKIPIF1<0；④對(duì)任意的點(diǎn)P，SKIPIF1<0的面積都不等于SKIPIF1<0.33．（2023·全國(guó)·高三專(zhuān)題練習(xí)）如圖所示，SKIPIF1<0是棱長(zhǎng)為SKIPIF1<0的正方體，SKIPIF1<0、SKIPIF1<0分別是下底面的棱SKIPIF1<0、SKIPIF1<0的中點(diǎn)，SKIPIF1<0是上底面的棱SKIPIF1<0上的一點(diǎn)，SKIPIF1<0，過(guò)SKIPIF1<0、SKIPIF1<0、SKIPIF1<0的平面交上底面于SKIPIF1<0，SKIPIF1<0在SKIPIF1<0上，則異面直線SKIPIF1<0與SKIPIF1<0所成角的余弦值為_(kāi)__________．34．（2022·河南鶴壁·鶴壁高中?？寄M預(yù)測(cè)）如圖，在棱長(zhǎng)為1的正方體SKIPIF1<0中，P為棱SKIPIF1<0的中點(diǎn)，Q為正方形SKIPIF1<0內(nèi)一動(dòng)點(diǎn)（含邊界），則下列說(shuō)法中正確的是______．①若SKIPIF1<0平面SKIPIF1<0，則動(dòng)點(diǎn)Q的軌跡是一條線段②存在Q點(diǎn)，使得SKIPIF1<0平面SKIPIF1<0③當(dāng)且僅當(dāng)Q點(diǎn)落在棱SKIPIF1<0上某點(diǎn)處時(shí)，三棱錐SKIPIF1<0的體積最大④若SKIPIF1<0，那么Q點(diǎn)的軌跡長(zhǎng)度為SKIPIF1<0四、解答題35．（2023·全國(guó)·高三專(zhuān)題練習(xí)）如圖，在四棱錐SKIPIF1<0中，SKIPIF1<0平面SKIPIF1<0，SKIPIF1<0，SKIPIF1<0，且SKIPIF1<0，SKIPIF1<0，SKIPIF1<0是SKIPIF1<0的中點(diǎn)，點(diǎn)SKIPIF1<0在SKIPIF1<0上，且SKIPIF1<0．(1)證明：SKIPIF1<0平面SKI