第06讲 空间向量及其运算的坐标表示（七大题型）（学生版）-2024年高中数学新高二暑期衔接讲义

第第页第06讲空间向量及其运算的坐标表示【题型归纳目录】题型一：空间向量的坐标表示题型二：空间向量的直角坐标运算题型三：空间向量的共线与共面题型四：空间向量模长坐标表示题型五：空间向量平行坐标表示题型六：空间向量垂直坐标表示题型七：空间向量夹角坐标表示【知识点梳理】知识点一、空间直角坐标系1、空间直角坐标系从空间某一定点O引三条互相垂直且有相同单位长度的数轴，这样就建立了空间直角坐标系SKIPIF1<0，点O叫做坐标原点，x轴、y轴、z轴叫做坐标轴，这三条坐标轴中每两条确定一个坐标平面，分别是SKIPIF1<0平面、yOz平面、zOx平面.2、右手直角坐标系在空间直角坐标系中，让右手拇指指向x轴的正方向，食指指向y轴的正方向，如果中指指向z轴的正方向，则称这个坐标系为右手直角坐标系.3、空间点的坐标空间一点A的坐标可以用有序数组(x，y，z)来表示，有序数组(x，y，z)叫做点A的坐标，记作A(x，y，z)，其中x叫做点A的横坐标，y叫做点A的纵坐标，z叫做点A的竖坐标.知识点二、空间直角坐标系中点的坐标1、空间直角坐标系中点的坐标的求法通过该点，作两条轴所确定平面的平行平面，此平面交另一轴于一点，交点在这条轴上的坐标就是已知点相应的一个坐标.特殊点的坐标：原点SKIPIF1<0；SKIPIF1<0轴上的点的坐标分别为SKIPIF1<0；坐标平面SKIPIF1<0上的点的坐标分别为SKIPIF1<0.2、空间直角坐标系中对称点的坐标在空间直角坐标系中，点SKIPIF1<0，则有点SKIPIF1<0关于原点的对称点是SKIPIF1<0；点SKIPIF1<0关于横轴(x轴)的对称点是SKIPIF1<0；点SKIPIF1<0关于纵轴(y轴)的对称点是SKIPIF1<0；点SKIPIF1<0关于竖轴(z轴)的对称点是SKIPIF1<0；点SKIPIF1<0关于坐标平面SKIPIF1<0的对称点是SKIPIF1<0；点SKIPIF1<0关于坐标平面SKIPIF1<0的对称点是SKIPIF1<0；点SKIPIF1<0关于坐标平面SKIPIF1<0的对称点是SKIPIF1<0.知识点三、空间向量的坐标运算(1)空间两点的距离公式若SKIPIF1<0，则①SKIPIF1<0即:一个向量在直角坐标系中的坐标等于表示这个向量的有向线段的终点的坐标减去起点的坐标。②SKIPIF1<0，或SKIPIF1<0.知识点诠释：两点间距离公式是模长公式的推广，首先根据向量的减法推出向量SKIPIF1<0的坐标表示，然后再用模长公式推出。(2)空间线段中点坐标空间中有两点SKIPIF1<0，则线段AB的中点C的坐标为SKIPIF1<0.(3)向量加减法、数乘的坐标运算若SKIPIF1<0，则①SKIPIF1<0;②SKIPIF1<0;③SKIPIF1<0;(4)向量数量积的坐标运算若SKIPIF1<0，则SKIPIF1<0即：空间两个向量的数量积等于他们的对应坐标的乘积之和。(5)空间向量长度及两向量夹角的坐标计算公式若SKIPIF1<0，则(1)SKIPIF1<0.(2)SKIPIF1<0.知识点诠释：①夹角公式可以根据数量积的定义推出：SKIPIF1<0，其中SKIPIF1<0的范围是SKIPIF1<0②SKIPIF1<0.③用此公式求异面直线所成角等角度时，要注意所求角度与θ的关系(相等，互余，互补)。(6)空间向量平行和垂直的条件若SKIPIF1<0，则①SKIPIF1<0②SKIPIF1<0规定：SKIPIF1<0与任意空间向量平行或垂直作用：证明线线平行、线线垂直.【典例例题】题型一：空间向量的坐标表示例1．在空间直角坐标系中，点SKIPIF1<0关于SKIPIF1<0轴对称的点坐标是(

)A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0例2．已知平行四边形SKIPIF1<0，且SKIPIF1<0，SKIPIF1<0，SKIPIF1<0，则顶点SKIPIF1<0的坐标为(

)A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<0例3．已知SKIPIF1<0，则向量SKIPIF1<0的坐标是(

)A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<0例4．平行六面体SKIPIF1<0中，SKIPIF1<0，则点SKIPIF1<0的坐标为(

)A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0例5．在空间直角坐标系中，SKIPIF1<0，SKIPIF1<0，则向量SKIPIF1<0(

)A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<0例6．若SKIPIF1<0､SKIPIF1<0，点C在线段AB上，且SKIPIF1<0，则点C的坐标是___________.例7．已知点SKIPIF1<0，SKIPIF1<0，SKIPIF1<0，则点SKIPIF1<0的坐标为______．例8．已知点SKIPIF1<0，SKIPIF1<0，若点SKIPIF1<0为线段AB上靠近SKIPIF1<0的三等分点，则点SKIPIF1<0的坐标为___________.例9．如图所示，在正方体ABCD—A1B1C1D1中建立空间直角坐标系，若正方体的棱长为1，则SKIPIF1<0的坐标为____，SKIPIF1<0的坐标为____，SKIPIF1<0的坐标为_______．题型二：空间向量的直角坐标运算例10．已知SKIPIF1<0，SKIPIF1<0，则SKIPIF1<0(

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

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

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

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

)A．SKIPIF1<0B．SKIPIF1<0C．SKIPIF1<0D．SKIPIF1<0例16．已知向量SKIPIF1<0则SKIPIF1<0的坐标为(

)A．SKIPIF1<0B．SKIPIF1<0C．SKIPIF1<0D．SKIPIF1<0题型三：空间向量的共线与共面例17．已知SKIPIF1<0，SKIPIF1<0，若SKIPIF1<0与SKIPIF1<0共线，则实数SKIPIF1<0(

)A．-2 B．SKIPIF1<0 C．SKIPIF1<0 D．2例18．与向量SKIPIF1<0共线的单位向量可以为(

)A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0例19．在空间直角坐标系中，已知点SKIPIF1<0，若SKIPIF1<0三点共线，则SKIPIF1<0(

)A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0例20．已知向量SKIPIF1<0，则与SKIPIF1<0同向共线的单位向量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为共线向量,则SKIPIF1<0的值为(

)A．2 B．SKIPIF1<0 C．6 D．8例22．若SKIPIF1<0三点共线，则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，则实数SKIPIF1<0的值为(

)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，则实数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，则实数SKIPIF1<0的值是(

)A．1 B．SKIPIF1<0 C．2 D．SKIPIF1<0例26．若空间四点SKIPIF1<0共面，则SKIPIF1<0的值为(

)A．SKIPIF1<0 B．2 C．1 D．SKIPIF1<0题型四：空间向量模长坐标表示例27．设空间向量SKIPIF1<0，SKIPIF1<0，若SKIPIF1<0，则SKIPIF1<0＝______．例28．SKIPIF1<0，SKIPIF1<0，则SKIPIF1<0_______．例29．已知SKIPIF1<0.则SKIPIF1<0__________．例30．已知点SKIPIF1<0是点SKIPIF1<0关于坐标平面yoz内的对称点，则SKIPIF1<0__________例31．若SKIPIF1<0，则SKIPIF1<0__________．例32．若SKIPIF1<0，则SKIPIF1<0___________题型五：空间向量平行坐标表示例33．在空间直角坐标系中，SKIPIF1<0，SKIPIF1<0，若SKIPIF1<0，则实数SKIPIF1<0__________.例34．已知向量SKIPIF1<0与向量SKIPIF1<0平行(SKIPIF1<0)，则SKIPIF1<0的值为______.例35．已知空间向量SKIPIF1<0，SKIPIF1<0，且SKIPIF1<0与SKIPIF1<0是共线向量，则实数x的值为_______．例36．已知空间向量SKIPIF1<0与SKIPIF1<0共线，则SKIPIF1<0______.题型六：空间向量垂直坐标表示例37．已知SKIPIF1<0，单位向量SKIPIF1<0满足SKIPIF1<0，则SKIPIF1<0_________．例38．已知空间向量SKIPIF1<0，SKIPIF1<0，若SKIPIF1<0，则SKIPIF1<0______.例39．已知向量SKIPIF1<0，SKIPIF1<0，若SKIPIF1<0，则SKIPIF1<0的值为_________.例40．已知向量SKIPIF1<0，且SKIPIF1<0与SKIPIF1<0互相垂直，则SKIPIF1<0的值是__．例41．已知空间向量SKIPIF1<0，若SKIPIF1<0，则x=___________.题型七：空间向量夹角坐标表示例42．已知向量SKIPIF1<0，SKIPIF1<0，若SKIPIF1<0与SKIPIF1<0的夹角为钝角，则实数SKIPIF1<0的取值范围为______.例43．在空间直角坐标系中，已知SKIPIF1<0，SKIPIF1<0，则SKIPIF1<0、SKIPIF1<0夹角的余弦值是______．例44．已知向量SKIPIF1<0，SKIPIF1<0，若SKIPIF1<0与SKIPIF1<0夹角为SKIPIF1<0，则k的值为________．例45．已知SKIPIF1<0、SKIPIF1<0、SKIPIF1<0，则向量SKIPIF1<0与SKIPIF1<0的夹角大小为______．例46．若向量SKIPIF1<0，SKIPIF1<0，且SKIPIF1<0与SKIPIF1<0的夹角的余弦值为SKIPIF1<0，则实数SKIPIF1<0的值为__________SKIPIF1<0例47．已知向量SKIPIF1<0，SKIPIF1<0，SKIPIF1<0，若向量SKIPIF1<0与SKIPIF1<0所成角为钝角，则实数SKIPIF1<0的范围是______.例48．已知向量SKIPIF1<0，若向量SKIPIF1<0与SKIPIF1<0的夹角为锐角，则实数SKIPIF1<0的取值范围是__________.【过关测试】一、单选题1．下列各组空间向量不能构成空间的一组基底的是(

)A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<02．若向量SKIPIF1<0，且SKIPIF1<0与SKIPIF1<0夹角的余弦值为SKIPIF1<0，则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，则SKIPIF1<0的值为(

)A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<04．已知向量SKIPIF1<0,若三个向量SKIPIF1<0共面,则实数m等于(

)A．4 B．6 C．8 D．105．长方体SKIPIF1<0中，SKIPIF1<0，E为SKIPIF1<0与SKIPIF1<0的交点，F为SKIPIF1<0与SKIPIF1<0的交点，又SKIPIF1<0，则长方体的高SKIPIF1<0等于(

)A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<06．设SKIPIF1<0，则AB的中点M到点C的距离SKIPIF1<0(

)A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<07．已知向量SKIPIF1<0，向量SKIPIF1<0，且平行四边形OACB对角线的中点坐标为SKIPIF1<0，则SKIPIF1<0(

)A．SKIPIF1<0 B．SKIPIF1<0C．SKIPIF1<0 D．SKIPIF1<08．已知空间直角坐标系SKIPIF1<0中，SKIPIF1<0，点SKIPIF1<0在直线SKIPIF1<0上运动，则当SKIPIF1<0取得最小值时，点SKIPIF1<0的坐标为(

)A．SKIPIF1<0 B．SKIPIF1<0 C．SKIPIF1<0 D．SKIPIF1<0二、多选题9．空间直角坐标系中，已知SKIPIF1<0，SKIPIF1<0，SKIPIF1<0，SKIPIF1<0，则(

)A．SKIPIF1<0B．SKIPIF1<0是等腰直角三角形C．与SKIPIF1<0平行的单位向量的坐标为SKIPIF1<0或SKIPIF1<0D．SKIPIF1<0在SKIPIF1<0方向上的投影向量的坐标为SKIPIF1<010．已知向量SKIPIF1<0，SKIPIF1<0，则下列结论中正确的是(

)A．若SKIPIF1<0，则SKIPIF1<0B．若SKIPIF1<0，则SKIPIF1<0C．不存在实数，使得SKIPIF1<0D．若SKIPIF1<0，则SKIPIF1<011．下列关于空间向量的命题中，正确的有(

)A．若向量SKIPIF1<0是空间的一个基底，则SKIPIF1<0也是空间的一个基底B．若SKIPIF1<0，则SKIPIF1<0的夹角是钝角C．已知SKIPIF1<0，SKIPIF1<0，若SKIPIF1<0与SKIPIF1<0垂直，则SKIPIF1<0D．已知A、B、C是空间中不共线的三个点，若点O满足SKIPIF1<0，则点O是唯一的，且一定与A、B、C共面12．如图，矩形ADFE、矩形CDFG、正方形ABCD两两垂直，且SKIPIF1<0，若线段DE上存在点P，使得SKIPIF1<0，则边CG长度的可能值为(

)A．2 B．SKIPIF1<0 C．4 D．SKIPIF1<0三、填空题13．已知点SKIPIF1<0，SKIPIF1<0，向量SKIPIF1<0，则点SKIPIF1<0的坐标为______．14．已知向量SKIPIF1<0，若向量SKIPIF1<0与SKIPIF1<0的夹角为锐角，求实数SKIPIF1<0的取值范围______.15．已知空间向量SKIPIF1<0，SKIPIF1<0，SKIPIF1<0，若SKIPIF1<0，SKIPIF1<0，SKIPIF1<0共面，则SKIPIF1<0______．16．已知SKIPIF1<0，SKIPIF1<0，且SKIPIF1<0，则SKIPIF1<0为______.四、解答题17．已知空间三点SKIPIF1<0，SKIPIF1<0，SKIPIF1<0，设SKIPIF1<0，SKIPIF1<0．(1)设SKIPIF1<0，SKIPIF1<0，求SKIPIF1<0；(2)求SKIPIF1<0与SKIPIF1<