第06講 空間向量及其運算的坐標(biāo)表示(七大題型)(學(xué)生版)-2024年高中數(shù)學(xué)新高二暑期銜接講義_第1頁
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第第頁第06講空間向量及其運算的坐標(biāo)表示【題型歸納目錄】題型一:空間向量的坐標(biāo)表示題型二:空間向量的直角坐標(biāo)運算題型三:空間向量的共線與共面題型四:空間向量模長坐標(biāo)表示題型五:空間向量平行坐標(biāo)表示題型六:空間向量垂直坐標(biāo)表示題型七:空間向量夾角坐標(biāo)表示【知識點梳理】知識點一、空間直角坐標(biāo)系1、空間直角坐標(biāo)系從空間某一定點O引三條互相垂直且有相同單位長度的數(shù)軸,這樣就建立了空間直角坐標(biāo)系SKIPIF1<0,點O叫做坐標(biāo)原點,x軸、y軸、z軸叫做坐標(biāo)軸,這三條坐標(biāo)軸中每兩條確定一個坐標(biāo)平面,分別是SKIPIF1<0平面、yOz平面、zOx平面.2、右手直角坐標(biāo)系在空間直角坐標(biāo)系中,讓右手拇指指向x軸的正方向,食指指向y軸的正方向,如果中指指向z軸的正方向,則稱這個坐標(biāo)系為右手直角坐標(biāo)系.3、空間點的坐標(biāo)空間一點A的坐標(biāo)可以用有序數(shù)組(x,y,z)來表示,有序數(shù)組(x,y,z)叫做點A的坐標(biāo),記作A(x,y,z),其中x叫做點A的橫坐標(biāo),y叫做點A的縱坐標(biāo),z叫做點A的豎坐標(biāo).知識點二、空間直角坐標(biāo)系中點的坐標(biāo)1、空間直角坐標(biāo)系中點的坐標(biāo)的求法通過該點,作兩條軸所確定平面的平行平面,此平面交另一軸于一點,交點在這條軸上的坐標(biāo)就是已知點相應(yīng)的一個坐標(biāo).特殊點的坐標(biāo):原點SKIPIF1<0;SKIPIF1<0軸上的點的坐標(biāo)分別為SKIPIF1<0;坐標(biāo)平面SKIPIF1<0上的點的坐標(biāo)分別為SKIPIF1<0.2、空間直角坐標(biāo)系中對稱點的坐標(biāo)在空間直角坐標(biāo)系中,點SKIPIF1<0,則有點SKIPIF1<0關(guān)于原點的對稱點是SKIPIF1<0;點SKIPIF1<0關(guān)于橫軸(x軸)的對稱點是SKIPIF1<0;點SKIPIF1<0關(guān)于縱軸(y軸)的對稱點是SKIPIF1<0;點SKIPIF1<0關(guān)于豎軸(z軸)的對稱點是SKIPIF1<0;點SKIPIF1<0關(guān)于坐標(biāo)平面SKIPIF1<0的對稱點是SKIPIF1<0;點SKIPIF1<0關(guān)于坐標(biāo)平面SKIPIF1<0的對稱點是SKIPIF1<0;點SKIPIF1<0關(guān)于坐標(biāo)平面SKIPIF1<0的對稱點是SKIPIF1<0.知識點三、空間向量的坐標(biāo)運算(1)空間兩點的距離公式若SKIPIF1<0,則①SKIPIF1<0即:一個向量在直角坐標(biāo)系中的坐標(biāo)等于表示這個向量的有向線段的終點的坐標(biāo)減去起點的坐標(biāo)。②SKIPIF1<0,或SKIPIF1<0.知識點詮釋:兩點間距離公式是模長公式的推廣,首先根據(jù)向量的減法推出向量SKIPIF1<0的坐標(biāo)表示,然后再用模長公式推出。(2)空間線段中點坐標(biāo)空間中有兩點SKIPIF1<0,則線段AB的中點C的坐標(biāo)為SKIPIF1<0.(3)向量加減法、數(shù)乘的坐標(biāo)運算若SKIPIF1<0,則①SKIPIF1<0;②SKIPIF1<0;③SKIPIF1<0;(4)向量數(shù)量積的坐標(biāo)運算若SKIPIF1<0,則SKIPIF1<0即:空間兩個向量的數(shù)量積等于他們的對應(yīng)坐標(biāo)的乘積之和。(5)空間向量長度及兩向量夾角的坐標(biāo)計算公式若SKIPIF1<0,則(1)SKIPIF1<0.(2)SKIPIF1<0.知識點詮釋:①夾角公式可以根據(jù)數(shù)量積的定義推出:SKIPIF1<0,其中SKIPIF1<0的范圍是SKIPIF1<0②SKIPIF1<0.③用此公式求異面直線所成角等角度時,要注意所求角度與θ的關(guān)系(相等,互余,互補)。(6)空間向量平行和垂直的條件若SKIPIF1<0,則①SKIPIF1<0②SKIPIF1<0規(guī)定:SKIPIF1<0與任意空間向量平行或垂直作用:證明線線平行、線線垂直.【典例例題】題型一:空間向量的坐標(biāo)表示例1.在空間直角坐標(biāo)系中,點SKIPIF1<0關(guān)于SKIPIF1<0軸對稱的點坐標(biāo)是(

)A.SKIPIF1<0 B.SKIPIF1<0 C.SKIPIF1<0 D.SKIPIF1<0例2.已知平行四邊形SKIPIF1<0,且SKIPIF1<0,SKIPIF1<0,SKIPIF1<0,則頂點SKIPIF1<0的坐標(biāo)為(

)A.SKIPIF1<0 B.SKIPIF1<0C.SKIPIF1<0 D.SKIPIF1<0例3.已知SKIPIF1<0,則向量SKIPIF1<0的坐標(biāo)是(

)A.SKIPIF1<0 B.SKIPIF1<0C.SKIPIF1<0 D.SKIPIF1<0例4.平行六面體SKIPIF1<0中,SKIPIF1<0,則點SKIPIF1<0的坐標(biāo)為(

)A.SKIPIF1<0 B.SKIPIF1<0 C.SKIPIF1<0 D.SKIPIF1<0例5.在空間直角坐標(biāo)系中,SKIPIF1<0,SKIPIF1<0,則向量SKIPIF1<0(

)A.SKIPIF1<0 B.SKIPIF1<0C.SKIPIF1<0 D.SKIPIF1<0例6.若SKIPIF1<0?SKIPIF1<0,點C在線段AB上,且SKIPIF1<0,則點C的坐標(biāo)是___________.例7.已知點SKIPIF1<0,SKIPIF1<0,SKIPIF1<0,則點SKIPIF1<0的坐標(biāo)為______.例8.已知點SKIPIF1<0,SKIPIF1<0,若點SKIPIF1<0為線段AB上靠近SKIPIF1<0的三等分點,則點SKIPIF1<0的坐標(biāo)為___________.例9.如圖所示,在正方體ABCD—A1B1C1D1中建立空間直角坐標(biāo)系,若正方體的棱長為1,則SKIPIF1<0的坐標(biāo)為____,SKIPIF1<0的坐標(biāo)為____,SKIPIF1<0的坐標(biāo)為_______.題型二:空間向量的直角坐標(biāo)運算例10.已知SKIPIF1<0,SKIPIF1<0,則SKIPIF1<0(

)A.-5 B.-7 C.3 D.SKIPIF1<0例11.若SKIPIF1<0,SKIPIF1<0,SKIPIF1<0,則SKIPIF1<0(

)A.-11 B.3 C.4 D.15例12.已知向量SKIPIF1<0,則SKIPIF1<0(

)A.SKIPIF1<0 B.SKIPIF1<0 C.SKIPIF1<0 D.SKIPIF1<0例13.已知SKIPIF1<0,SKIPIF1<0,則SKIPIF1<0(

)A.SKIPIF1<0 B.SKIPIF1<0 C.SKIPIF1<0 D.SKIPIF1<0例14.已知向量SKIPIF1<0,SKIPIF1<0,若SKIPIF1<0,則SKIPIF1<0(

)A.SKIPIF1<0 B.SKIPIF1<0 C.SKIPIF1<0 D.SKIPIF1<0例15.已知SKIPIF1<0,SKIPIF1<0,則SKIPIF1<0等于(

)A.SKIPIF1<0B.SKIPIF1<0C.SKIPIF1<0D.SKIPIF1<0例16.已知向量SKIPIF1<0則SKIPIF1<0的坐標(biāo)為(

)A.SKIPIF1<0B.SKIPIF1<0C.SKIPIF1<0D.SKIPIF1<0題型三:空間向量的共線與共面例17.已知SKIPIF1<0,SKIPIF1<0,若SKIPIF1<0與SKIPIF1<0共線,則實數(shù)SKIPIF1<0(

)A.-2 B.SKIPIF1<0 C.SKIPIF1<0 D.2例18.與向量SKIPIF1<0共線的單位向量可以為(

)A.SKIPIF1<0 B.SKIPIF1<0 C.SKIPIF1<0 D.SKIPIF1<0例19.在空間直角坐標(biāo)系中,已知點SKIPIF1<0,若SKIPIF1<0三點共線,則SKIPIF1<0(

)A.SKIPIF1<0 B.SKIPIF1<0 C.SKIPIF1<0 D.SKIPIF1<0例20.已知向量SKIPIF1<0,則與SKIPIF1<0同向共線的單位向量SKIPIF1<0(

)A.SKIPIF1<0 B.SKIPIF1<0C.SKIPIF1<0 D.SKIPIF1<0例21.若SKIPIF1<0,SKIPIF1<0,且SKIPIF1<0,SKIPIF1<0為共線向量,則SKIPIF1<0的值為(

)A.2 B.SKIPIF1<0 C.6 D.8例22.若SKIPIF1<0三點共線,則SKIPIF1<0(

)A.1 B.4 C.6 D.2例23.已知向量SKIPIF1<0,SKIPIF1<0,SKIPIF1<0,若SKIPIF1<0共面,則實數(shù)SKIPIF1<0的值為(

)A.SKIPIF1<0 B.1 C.SKIPIF1<0 D.SKIPIF1<0例24.已知SKIPIF1<0,SKIPIF1<0,SKIPIF1<0,若SKIPIF1<0三向量共面,則實數(shù)SKIPIF1<0等于(

)A.SKIPIF1<0 B.SKIPIF1<0 C.SKIPIF1<0 D.SKIPIF1<0例25.已知向量SKIPIF1<0共面,則實數(shù)SKIPIF1<0的值是(

)A.1 B.SKIPIF1<0 C.2 D.SKIPIF1<0例26.若空間四點SKIPIF1<0共面,則SKIPIF1<0的值為(

)A.SKIPIF1<0 B.2 C.1 D.SKIPIF1<0題型四:空間向量模長坐標(biāo)表示例27.設(shè)空間向量SKIPIF1<0,SKIPIF1<0,若SKIPIF1<0,則SKIPIF1<0=______.例28.SKIPIF1<0,SKIPIF1<0,則SKIPIF1<0_______.例29.已知SKIPIF1<0.則SKIPIF1<0__________.例30.已知點SKIPIF1<0是點SKIPIF1<0關(guān)于坐標(biāo)平面yoz內(nèi)的對稱點,則SKIPIF1<0__________例31.若SKIPIF1<0,則SKIPIF1<0__________.例32.若SKIPIF1<0,則SKIPIF1<0___________題型五:空間向量平行坐標(biāo)表示例33.在空間直角坐標(biāo)系中,SKIPIF1<0,SKIPIF1<0,若SKIPIF1<0,則實數(shù)SKIPIF1<0__________.例34.已知向量SKIPIF1<0與向量SKIPIF1<0平行(SKIPIF1<0),則SKIPIF1<0的值為______.例35.已知空間向量SKIPIF1<0,SKIPIF1<0,且SKIPIF1<0與SKIPIF1<0是共線向量,則實數(shù)x的值為_______.例36.已知空間向量SKIPIF1<0與SKIPIF1<0共線,則SKIPIF1<0______.題型六:空間向量垂直坐標(biāo)表示例37.已知SKIPIF1<0,單位向量SKIPIF1<0滿足SKIPIF1<0,則SKIPIF1<0_________.例38.已知空間向量SKIPIF1<0,SKIPIF1<0,若SKIPIF1<0,則SKIPIF1<0______.例39.已知向量SKIPIF1<0,SKIPIF1<0,若SKIPIF1<0,則SKIPIF1<0的值為_________.例40.已知向量SKIPIF1<0,且SKIPIF1<0與SKIPIF1<0互相垂直,則SKIPIF1<0的值是__.例41.已知空間向量SKIPIF1<0,若SKIPIF1<0,則x=___________.題型七:空間向量夾角坐標(biāo)表示例42.已知向量SKIPIF1<0,SKIPIF1<0,若SKIPIF1<0與SKIPIF1<0的夾角為鈍角,則實數(shù)SKIPIF1<0的取值范圍為______.例43.在空間直角坐標(biāo)系中,已知SKIPIF1<0,SKIPIF1<0,則SKIPIF1<0、SKIPIF1<0夾角的余弦值是______.例44.已知向量SKIPIF1<0,SKIPIF1<0,若SKIPIF1<0與SKIPIF1<0夾角為SKIPIF1<0,則k的值為________.例45.已知SKIPIF1<0、SKIPIF1<0、SKIPIF1<0,則向量SKIPIF1<0與SKIPIF1<0的夾角大小為______.例46.若向量SKIPIF1<0,SKIPIF1<0,且SKIPIF1<0與SKIPIF1<0的夾角的余弦值為SKIPIF1<0,則實數(shù)SKIPIF1<0的值為__________SKIPIF1<0例47.已知向量SKIPIF1<0,SKIPIF1<0,SKIPIF1<0,若向量SKIPIF1<0與SKIPIF1<0所成角為鈍角,則實數(shù)SKIPIF1<0的范圍是______.例48.已知向量SKIPIF1<0,若向量SKIPIF1<0與SKIPIF1<0的夾角為銳角,則實數(shù)SKIPIF1<0的取值范圍是__________.【過關(guān)測試】一、單選題1.下列各組空間向量不能構(gòu)成空間的一組基底的是(

)A.SKIPIF1<0 B.SKIPIF1<0C.SKIPIF1<0 D.SKIPIF1<02.若向量SKIPIF1<0,且SKIPIF1<0與SKIPIF1<0夾角的余弦值為SKIPIF1<0,則SKIPIF1<0等于(

)A.SKIPIF1<0 B.SKIPIF1<0 C.SKIPIF1<0或SKIPIF1<0 D.23.已知向量SKIPIF1<0,若SKIPIF1<0,則SKIPIF1<0的值為(

)A.SKIPIF1<0 B.SKIPIF1<0 C.SKIPIF1<0 D.SKIPIF1<04.已知向量SKIPIF1<0,若三個向量SKIPIF1<0共面,則實數(shù)m等于(

)A.4 B.6 C.8 D.105.長方體SKIPIF1<0中,SKIPIF1<0,E為SKIPIF1<0與SKIPIF1<0的交點,F(xiàn)為SKIPIF1<0與SKIPIF1<0的交點,又SKIPIF1<0,則長方體的高SKIPIF1<0等于(

)A.SKIPIF1<0 B.SKIPIF1<0 C.SKIPIF1<0 D.SKIPIF1<06.設(shè)SKIPIF1<0,則AB的中點M到點C的距離SKIPIF1<0(

)A.SKIPIF1<0 B.SKIPIF1<0 C.SKIPIF1<0 D.SKIPIF1<07.已知向量SKIPIF1<0,向量SKIPIF1<0,且平行四邊形OACB對角線的中點坐標(biāo)為SKIPIF1<0,則SKIPIF1<0(

)A.SKIPIF1<0 B.SKIPIF1<0C.SKIPIF1<0 D.SKIPIF1<08.已知空間直角坐標(biāo)系SKIPIF1<0中,SKIPIF1<0,點SKIPIF1<0在直線SKIPIF1<0上運動,則當(dāng)SKIPIF1<0取得最小值時,點SKIPIF1<0的坐標(biāo)為(

)A.SKIPIF1<0 B.SKIPIF1<0 C.SKIPIF1<0 D.SKIPIF1<0二、多選題9.空間直角坐標(biāo)系中,已知SKIPIF1<0,SKIPIF1<0,SKIPIF1<0,SKIPIF1<0,則(

)A.SKIPIF1<0B.SKIPIF1<0是等腰直角三角形C.與SKIPIF1<0平行的單位向量的坐標(biāo)為SKIPIF1<0或SKIPIF1<0D.SKIPIF1<0在SKIPIF1<0方向上的投影向量的坐標(biāo)為SKIPIF1<010.已知向量SKIPIF1<0,SKIPIF1<0,則下列結(jié)論中正確的是(

)A.若SKIPIF1<0,則SKIPIF1<0B.若SKIPIF1<0,則SKIPIF1<0C.不存在實數(shù),使得SKIPIF1<0D.若SKIPIF1<0,則SKIPIF1<011.下列關(guān)于空間向量的命題中,正確的有(

)A.若向量SKIPIF1<0是空間的一個基底,則SKIPIF1<0也是空間的一個基底B.若SKIPIF1<0,則SKIPIF1<0的夾角是鈍角C.已知SKIPIF1<0,SKIPIF1<0,若SKIPIF1<0與SKIPIF1<0垂直,則SKIPIF1<0D.已知A、B、C是空間中不共線的三個點,若點O滿足SKIPIF1<0,則點O是唯一的,且一定與A、B、C共面12.如圖,矩形ADFE、矩形CDFG、正方形ABCD兩兩垂直,且SKIPIF1<0,若線段DE上存在點P,使得SKIPIF1<0,則邊CG長度的可能值為(

)A.2 B.SKIPIF1<0 C.4 D.SKIPIF1<0三、填空題13.已知點SKIPIF1<0,SKIPIF1<0,向量SKIPIF1<0,則點SKIPIF1<0的坐標(biāo)為______.14.已知向量SKIPIF1<0,若向量SKIPIF1<0與SKIPIF1<0的夾角為銳角,求實數(shù)SKIPIF1<0的取值范圍______.15.已知空間向量SKIPIF1<0,SKIPIF1<0,SKIPIF1<0,若SKIPIF1<0,SKIPIF1<0,SKIPIF1<0共面,則SKIPIF1<0______.16.已知SKIPIF1<0,SKIPIF1<0,且SKIPIF1<0,則SKIPIF1<0為______.四、解答題17.已知空間三點SKIPIF1<0,SKIPIF1<0,SKIPIF1<0,設(shè)SKIPIF1<0,SKIPIF1<0.(1)設(shè)SKIPIF1<0,SKIPIF1<0,求SKIPIF1<0;(2)求SKIPIF1<0與SKIPIF1<

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