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專題1.7空間向量與立體幾何(六個(gè)混淆易錯(cuò)點(diǎn))易錯(cuò)點(diǎn)1對(duì)空間向量的運(yùn)算理解不清1.在棱長(zhǎng)為1的正四面體SKIPIF1<0中,點(diǎn)SKIPIF1<0滿足SKIPIF1<0,點(diǎn)SKIPIF1<0滿足SKIPIF1<0,當(dāng)線段SKIPIF1<0、SKIPIF1<0的長(zhǎng)度均最短時(shí),SKIPIF1<0(
)A.SKIPIF1<0 B.SKIPIF1<0 C.SKIPIF1<0 D.SKIPIF1<02.下列命題中正確的個(gè)數(shù)是(
).①若SKIPIF1<0與SKIPIF1<0共線,SKIPIF1<0與SKIPIF1<0共線,則SKIPIF1<0與SKIPIF1<0共線.②向量SKIPIF1<0,SKIPIF1<0,SKIPIF1<0共面,即它們所在的直線共面.③如果三個(gè)向量SKIPIF1<0,SKIPIF1<0,SKIPIF1<0不共面,那么對(duì)于空間任意一個(gè)向量SKIPIF1<0,存在有序?qū)崝?shù)組SKIPIF1<0,使得SKIPIF1<0.④若SKIPIF1<0,SKIPIF1<0是兩個(gè)不共線的向量,而SKIPIF1<0(SKIPIF1<0且SKIPIF1<0),則SKIPIF1<0是空間向量的一組基底.A.0 B.1 C.2 D.33.以下命題:①若SKIPIF1<0,則存在唯一的實(shí)數(shù)SKIPIF1<0,使得SKIPIF1<0;②若SKIPIF1<0,則SKIPIF1<0或SKIPIF1<0;③若SKIPIF1<0為空間的一個(gè)基底,則SKIPIF1<0構(gòu)成空間的另一個(gè)基底;④SKIPIF1<0一定成立.則其中真命題的個(gè)數(shù)為(
)A.4 B.3 C.2 D.14.下面四個(gè)結(jié)論正確的個(gè)數(shù)是(
)①空間向量SKIPIF1<0,若SKIPIF1<0,則SKIPIF1<0;②若空間四個(gè)點(diǎn)P,A,B,C,SKIPIF1<0,則A,B,C三點(diǎn)共線;③已知向量SKIPIF1<0,SKIPIF1<0,若SKIPIF1<0,則SKIPIF1<0為鈍角;④任意向量SKIPIF1<0滿足SKIPIF1<0.A.4 B.3 C.2 D.15.(多選)給出下列命題,其中正確的是()A.若SKIPIF1<0是空間的一個(gè)基底,則SKIPIF1<0也是空間的一個(gè)基底B.在空間直角坐標(biāo)系中,點(diǎn)SKIPIF1<0關(guān)于坐標(biāo)平面yOz的對(duì)稱點(diǎn)是SKIPIF1<0C.若空間四個(gè)點(diǎn)P,A,B,C滿足SKIPIF1<0,則A,B,C三點(diǎn)共線D.平面SKIPIF1<0的一個(gè)法向量為SKIPIF1<0,平面SKIPIF1<0的一個(gè)法向量為SKIPIF1<0.若SKIPIF1<0,則SKIPIF1<06.(多選)下列命題中正確的是(
)A.SKIPIF1<0是SKIPIF1<0,SKIPIF1<0共線的充分條件B.若SKIPIF1<0,則SKIPIF1<0C.SKIPIF1<0,SKIPIF1<0,SKIPIF1<0三點(diǎn)不共線,對(duì)空間任意一點(diǎn)SKIPIF1<0,若SKIPIF1<0,則SKIPIF1<0,SKIPIF1<0,SKIPIF1<0,SKIPIF1<0四點(diǎn)共面D.若SKIPIF1<0,SKIPIF1<0,SKIPIF1<0,SKIPIF1<0為空間四點(diǎn),且有SKIPIF1<0(SKIPIF1<0,SKIPIF1<0不共線),則SKIPIF1<0是SKIPIF1<0,SKIPIF1<0,SKIPIF1<0三點(diǎn)共線的充分不必要條件7.在正四面體SKIPIF1<0中,SKIPIF1<0,SKIPIF1<0,SKIPIF1<0,SKIPIF1<0分別是SKIPIF1<0,SKIPIF1<0,SKIPIF1<0,SKIPIF1<0的中點(diǎn).設(shè)SKIPIF1<0,SKIPIF1<0,SKIPIF1<0.(1)用SKIPIF1<0,SKIPIF1<0,SKIPIF1<0表示SKIPIF1<0,SKIPIF1<0;(2)求證:SKIPIF1<0;(3)求證:SKIPIF1<0,SKIPIF1<0,SKIPIF1<0,SKIPIF1<0四點(diǎn)共面.易錯(cuò)點(diǎn)2忽略條件導(dǎo)致建系錯(cuò)誤8.如圖,在直四棱柱SKIPIF1<0中,SKIPIF1<0,SKIPIF1<0,SKIPIF1<0,SKIPIF1<0.點(diǎn)SKIPIF1<0在棱SKIPIF1<0上,平面SKIPIF1<0與棱SKIPIF1<0交于點(diǎn)SKIPIF1<0.(1)求證:SKIPIF1<0;(2)若SKIPIF1<0與平面SKIPIF1<0所成角的正弦值為SKIPIF1<0,試確定點(diǎn)SKIPIF1<0的位置.9.如圖,在三棱柱SKIPIF1<0中,SKIPIF1<0平面SKIPIF1<0,SKIPIF1<0,SKIPIF1<0,SKIPIF1<0,SKIPIF1<0分別為SKIPIF1<0,SKIPIF1<0,SKIPIF1<0,SKIPIF1<0的中點(diǎn),SKIPIF1<0,SKIPIF1<0.
(1)試建立空間直角坐標(biāo)系,并寫出點(diǎn)SKIPIF1<0,SKIPIF1<0的坐標(biāo);(2)求SKIPIF1<0的余弦值.10.如圖所示,正三棱柱SKIPIF1<0的所有棱長(zhǎng)都為SKIPIF1<0,SKIPIF1<0為SKIPIF1<0的中點(diǎn).請(qǐng)建立適當(dāng)空間直角坐標(biāo)系,并求各個(gè)點(diǎn)的坐標(biāo).11.如圖,在三棱柱SKIPIF1<0中,SKIPIF1<0側(cè)面SKIPIF1<0,SKIPIF1<0為棱SKIPIF1<0上異于SKIPIF1<0的一點(diǎn),SKIPIF1<0.已知SKIPIF1<0,SKIPIF1<0,SKIPIF1<0.請(qǐng)建立適當(dāng)空間直角坐標(biāo)系,并求各個(gè)點(diǎn)的坐標(biāo).12.在平行六面體SKIPIF1<0中,底面SKIPIF1<0是矩形,SKIPIF1<0,SKIPIF1<0,平行六面體高為SKIPIF1<0,頂點(diǎn)SKIPIF1<0在底面SKIPIF1<0的射影SKIPIF1<0是SKIPIF1<0中點(diǎn),設(shè)SKIPIF1<0的重心SKIPIF1<0,建立適當(dāng)空間直角坐標(biāo)系并寫出點(diǎn)SKIPIF1<0的坐標(biāo).SKIPIF1<013.如圖所示,已知平行六面體SKIPIF1<0的底面為邊長(zhǎng)為SKIPIF1<0的正方形,SKIPIF1<0分別為上、下底面的中心,且SKIPIF1<0在底面SKIPIF1<0上的射影是SKIPIF1<0,且SKIPIF1<0.請(qǐng)建立適當(dāng)空間直角坐標(biāo)系,并求點(diǎn)SKIPIF1<0的坐標(biāo).14.如圖,在三棱柱SKIPIF1<0中,平面SKIPIF1<0平面SKIPIF1<0,SKIPIF1<0,SKIPIF1<0且SKIPIF1<0,SKIPIF1<0,請(qǐng)建立適當(dāng)空間直角坐標(biāo)系,并求各個(gè)點(diǎn)的坐標(biāo).易錯(cuò)點(diǎn)3證明線面平行垂直時(shí)出現(xiàn)混亂15.設(shè)直線SKIPIF1<0的方向向量為SKIPIF1<0,平面SKIPIF1<0的法向量為SKIPIF1<0,若SKIPIF1<0,則(
)A.SKIPIF1<0 B.SKIPIF1<0 C.SKIPIF1<0 D.SKIPIF1<0或SKIPIF1<016.設(shè)直線SKIPIF1<0的方向向量為SKIPIF1<0,SKIPIF1<0,SKIPIF1<0,SKIPIF1<0為平面SKIPIF1<0的三點(diǎn),則直線SKIPIF1<0與平面SKIPIF1<0的位置關(guān)系是(
)A.SKIPIF1<0 B.SKIPIF1<0 C.SKIPIF1<0 D.SKIPIF1<0或SKIPIF1<017.(多選)在棱長(zhǎng)為1的正方體SKIPIF1<0中,E,F(xiàn)分別是AB,BC中點(diǎn),則(
)A.SKIPIF1<0平面SKIPIF1<0B.SKIPIF1<0平面SKIPIF1<0C.平面SKIPIF1<0平面SKIPIF1<0D.點(diǎn)E到平面SKIPIF1<0的距離為SKIPIF1<018.如圖,在四棱錐SKIPIF1<0中,底面ABCD是邊長(zhǎng)為1的正方形,SKIPIF1<0底面ABCD,垂足為A,SKIPIF1<0,點(diǎn)M在棱PD上,SKIPIF1<0平面ACM.
(1)試確定點(diǎn)M的位置;(2)計(jì)算直線PB與平面MAC的距離;(3)設(shè)點(diǎn)E在棱PC上,當(dāng)點(diǎn)E在何處時(shí),使得SKIPIF1<0平面PBD?19.已知在長(zhǎng)方體SKIPIF1<0中,SKIPIF1<0,SKIPIF1<0,在線段SKIPIF1<0上取點(diǎn)M,在SKIPIF1<0上取點(diǎn)N,使得直線SKIPIF1<0平面SKIPIF1<0,則線段MN長(zhǎng)度的最小值為(
)A.SKIPIF1<0 B.SKIPIF1<0 C.SKIPIF1<0 D.SKIPIF1<020.(多選)若SKIPIF1<0是平面SKIPIF1<0的一個(gè)法向量,SKIPIF1<0是平面SKIPIF1<0的一個(gè)法向量,SKIPIF1<0,SKIPIF1<0是直線SKIPIF1<0上不同的兩點(diǎn),則以下命題正確的是(
)A.SKIPIF1<0B.SKIPIF1<0C.SKIPIF1<0,使得SKIPIF1<0D.設(shè)SKIPIF1<0與SKIPIF1<0的夾角為SKIPIF1<0,則SKIPIF1<021.(多選)如圖,在正四棱柱SKIPIF1<0中,SKIPIF1<0,SKIPIF1<0,點(diǎn)SKIPIF1<0在棱SKIPIF1<0上,且SKIPIF1<0,點(diǎn)SKIPIF1<0在上底面SKIPIF1<0運(yùn)動(dòng),則下列結(jié)論正確的是(
)
A.存在點(diǎn)SKIPIF1<0使SKIPIF1<0B.不存在點(diǎn)SKIPIF1<0使平面SKIPIF1<0平面SKIPIF1<0C.若SKIPIF1<0,SKIPIF1<0,SKIPIF1<0,SKIPIF1<0四點(diǎn)共面,則SKIPIF1<0的最小值為SKIPIF1<0D.若SKIPIF1<0,SKIPIF1<0,SKIPIF1<0,SKIPIF1<0,SKIPIF1<0五點(diǎn)共球面,則SKIPIF1<0的最小值為SKIPIF1<022.如圖所示的幾何體中,平面SKIPIF1<0平面SKIPIF1<0為等腰直角三角形,SKIPIF1<0,四邊形SKIPIF1<0為直角梯形,SKIPIF1<0.
(1)求證:SKIPIF1<0SKIPIF1<0平面SKIPIF1<0;(2)線段SKIPIF1<0上是否存在點(diǎn)SKIPIF1<0滿足SKIPIF1<0,使得SKIPIF1<0平面SKIPIF1<0?若存在,求出SKIPIF1<0的值;若不存在,說明理由.易錯(cuò)點(diǎn)4混淆異面直線的夾角與向量的夾角23.已知直線SKIPIF1<0的方向向量為SKIPIF1<0,直線SKIPIF1<0的方向向量為SKIPIF1<0,則直線SKIPIF1<0與SKIPIF1<0所成角的度數(shù)為(
)A.SKIPIF1<0 B.SKIPIF1<0 C.SKIPIF1<0 D.SKIPIF1<024.如圖,在四棱錐SKIPIF1<0中,SKIPIF1<0底面SKIPIF1<0,底面SKIPIF1<0為正方形,SKIPIF1<0,SKIPIF1<0為SKIPIF1<0上一點(diǎn),且SKIPIF1<0,則異面直線SKIPIF1<0與SKIPIF1<0所成的角的大小為(
)A.SKIPIF1<0 B.SKIPIF1<0 C.SKIPIF1<0 D.SKIPIF1<025.(多選)在三棱錐SKIPIF1<0中,平面SKIPIF1<0平面BCD,SKIPIF1<0,SKIPIF1<0,SKIPIF1<0為等邊三角形,E是棱AC的中點(diǎn),F(xiàn)是棱AD上一點(diǎn),若異面直線DE與BF所成角的余弦值為SKIPIF1<0,則AF的值可能為(
)A.SKIPIF1<0 B.1 C.SKIPIF1<0 D.SKIPIF1<026.如圖,已知空間四邊形SKIPIF1<0的每條邊和對(duì)角線的長(zhǎng)都等于1,點(diǎn)SKIPIF1<0分別是SKIPIF1<0的中點(diǎn),計(jì)算:(1)SKIPIF1<0;(2)異面直線SKIPIF1<0與SKIPIF1<0所成角的余弦值.27.如圖,在正方體SKIPIF1<0中,點(diǎn)SKIPIF1<0分別是SKIPIF1<0的中點(diǎn),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<028.如圖在平行六面體SKIPIF1<0中,SKIPIF1<0,SKIPIF1<0,SKIPIF1<0,SKIPIF1<0,SKIPIF1<0?SKIPIF1<0?SKIPIF1<0分別為SKIPIF1<0,SKIPIF1<0,SKIPIF1<0的中點(diǎn).(1)求證:SKIPIF1<0(2)求SKIPIF1<0和SKIPIF1<0所成角的余弦值;29.如圖,在三棱柱SKIPIF1<0中,SKIPIF1<0平面SKIPIF1<0,SKIPIF1<0,SKIPIF1<0,SKIPIF1<0,SKIPIF1<0分別為SKIPIF1<0,SKIPIF1<0的中點(diǎn).(1)求證:SKIPIF1<0;(2)求異面直線SKIPIF1<0與SKIPIF1<0所成角的余弦值.30.設(shè)SKIPIF1<0分別是空間兩直線SKIPIF1<0的方向向量,則直線SKIPIF1<0,SKIPIF1<0所成角的大小為_____.易錯(cuò)點(diǎn)5混淆線面的夾角與向量的夾角31.如圖,在四棱錐SKIPIF1<0中,SKIPIF1<0平面SKIPIF1<0,SKIPIF1<0,SKIPIF1<0,SKIPIF1<0,已知Q是棱SKIPIF1<0上靠近點(diǎn)P的四等分點(diǎn),則SKIPIF1<0與平面SKIPIF1<0所成角的正弦值為(
).A.SKIPIF1<0 B.SKIPIF1<0 C.SKIPIF1<0 D.SKIPIF1<032.在直三棱柱SKIPIF1<0中,底面是等腰直角三角形,SKIPIF1<0,側(cè)棱SKIPIF1<0,D,E分別是SKIPIF1<0與SKIPIF1<0的中點(diǎn),點(diǎn)E在平面ABD上的射影是SKIPIF1<0的重心G,則SKIPIF1<0與平面ABD所成角的余弦(
)A.SKIPIF1<0 B.SKIPIF1<0 C.SKIPIF1<0 D.SKIPIF1<033.如圖,已知ABCD和CDEF都是直角梯形,SKIPIF1<0,SKIPIF1<0,SKIPIF1<0,二面角SKIPIF1<0的平面角為SKIPIF1<0.設(shè)M,N分別為AE,BC的中點(diǎn),直線BM與平面ADE所成角的正弦值為_____.
34.如圖,已知四邊形ABCD,CDGF,ADGE均為正方形,且邊長(zhǎng)為1,在棱DG上是否存在點(diǎn)M,使得直線MB與平面BEF所成的角為45°?若存在,求出點(diǎn)M的位置;若不存在,試說明理由.35.在正四棱柱SKIPIF1<0中,SKIPIF1<0,E為SKIPIF1<0的中點(diǎn).(用向量的方法證明)(1)求證:SKIPIF1<0平面SKIPIF1<0.(用向量的方法證明)(2)若F為SKIPIF1<0上的動(dòng)點(diǎn),使直線SKIPIF1<0與平面SKIPIF1<0所成角的正弦值是SKIPIF1<0,求BF的長(zhǎng).36.如圖1,已知正方形ABCD的邊長(zhǎng)為4,E,F(xiàn)分別為AD,BC的中點(diǎn),將正方形ABCD沿EF折成如圖2所示的二面角,點(diǎn)M在線段AB上(含端點(diǎn))運(yùn)動(dòng),連接AD.(1)若M為AB的中點(diǎn),直線MF與平面ADE交于點(diǎn)O,確定O點(diǎn)位置,求線段OA的長(zhǎng);(2)若折成二面角的大小為45°,是否存在點(diǎn)M,使得直線DE與平面EMC所成的角為45°,若存在,確定出點(diǎn)M的位置;若不存在,請(qǐng)說明理由.37.如圖,在四棱錐SKIPIF1<0中,SKIPIF1<0平面ABCD,底面ABCD是正方形,點(diǎn)E在棱PD上,SKIPIF1<0,SKIPIF1<0.(1)證明:點(diǎn)E是PD的中點(diǎn);(2)求直線BE與平面ACE所成角的余弦值.38.已知多面體SKIPIF1<0,四邊形SKIPIF1<0是等腰梯形,SKIPIF1<0,SKIPIF1<0,四邊形SKIPIF1<0是菱形,SKIPIF1<0,E,F(xiàn)分別為QA,BC的中點(diǎn),SKIPIF1<0.
(1)求證:平面SKIPIF1<0平面SKIPIF1<0;(2)求直線SKIPIF1<0與平面SKIPIF1<0夾角的正弦值.易錯(cuò)點(diǎn)6混淆兩個(gè)平面的夾角與二面角39.在正六棱柱SKIPIF1<0中,底面棱長(zhǎng)為SKIPIF1<0,高為SKIPIF1<0,SKIPIF1<0分別為SKIPIF1<0,SKIPIF1<0的中點(diǎn),連接SKIPIF1<0.
(1)求SKIPIF1<0所成角SKIPIF1<0的余弦值;(2)過點(diǎn)SKIPIF1<0作直線SKIPIF1<0,設(shè)點(diǎn)SKIPIF1<0是直線SKIPIF1<0上一點(diǎn),記平面SKIPIF1<0與平面SKIPIF1<0所成角為SKIPIF1<0,求SKIPIF1<0的取值范圍.40.如圖,在三棱柱SKIPIF1<0中,底面ABC是邊長(zhǎng)為8的等邊三角形,SKIPIF1<0,SKIPIF1<0,SKIPIF1<0,D在SKIPIF1<0上且滿足SKIPIF1<0.
(1)求證:平面SKIPIF1<0平面SKIPIF1<0;(2)求平面SKIPIF1<0與平面SKIPIF1<0夾角的正弦值.41.如圖1,在四邊形SKIPIF1<0中,SKIPIF1<0,SKIPIF1<0為SKIPIF1<0上一點(diǎn),SKIPIF1<0,SKIPIF1<0,SKIPIF1<0,將四邊形SKIPIF1<0沿SKIPIF1<0折起,使得二面角SKIPIF1<0的大小為SKIPIF1<0,連接SKIPIF1<0,SKIPIF1<0,得到如圖2.
(1)證明:平面SKIPIF1<0SKIPIF1<0平面SKIPIF1<0;(2)點(diǎn)SKIPIF1<0是線段SKIPIF1<0上一點(diǎn),設(shè)SKIPIF1<0,且二面角SKIPIF1<0為SKIPIF1<0,求SKIPIF1<0的值.42.如圖,四棱錐SKIPIF1<0的底面是矩形,側(cè)面SKIPIF1<0是正三角形,且側(cè)面SKIPIF1<0底面SKIPIF1<0,SKIPIF1<0為側(cè)棱SKIPIF1<0的中點(diǎn).
(1)求證:SKIPIF1<0平面SKIPIF1<0;(2)若SKI
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