Question

立体几何旋转问题,要指导结果1、 空间三个点P1(x1,y1)、P2(x2,y2)、P3(x3,y3),现在要把P3点绕

Answer 1

.2.)解:在空间中建立坐标系 O - XYZ ．现以 O 为始点的 三个单位向量 i (1,0,0), j (0,1,0),k (0,0,1) 为坐标基向量 ,对于Pi=( xi, yi, zi) 依次称 xii , yij , zik 三个向量为向量 P=( xi, yi, zi)关于 x 轴、 y 轴和 z 轴的分量则Pi=xii +yij +,zik 设P5P6 P7平面上的法向量为 N1=P7P6×P7P5单位法向量为N=(Nx, Ny, Nz) 则 向量叉积:N1 =P7P6×P7P5 =|N1|N N=Nxi + Nyj + Nz k则Nxi=[ (y6-y7 )(z5-z7)- (z6-z7)(y5-y7 ) ] /|N1|iNyj= - [ (x6-x7 )(z5-z7)- (z6-z7)(x5-x7 ) ] /|N1|jNzk=[ (x6-x7 )(y5-y7)- (y6-y7 )(x5-x7 )] /|N1|k ∴|N1| =√{[ (y6-y7 )(z5-z7)-(z6-z7)(y5-y7 )]2 +[ (x6-x7 )(z5-z7)- (z6-z7)(x5-x7 ) ] 2+[ (x6-x7 )(y5-y7 ) - (y6-y7 )(x5-x7 )] 2}∴平面 P5P6 P7点法式方程:Nx( x-x5 )+Ny( y-y5 ) + Nz( z-z5 )=0另过 P7 (x7, y7, z7) 点作直线 P5P6 的垂线,垂足为 D, P7到直线 P5P6的距离为|P7D|∵P7D ⊥P5P6∴2×△P7P5P6的面积=|P7D||P5P6|=|P7P6||P7P5|sin(P7P6&P7P5交角)=|P7P6×P7P5|∴|N1|=|P7P6×P7P5|=|P7D||P5P6|…①∴|P7D|=|N1|/|P5P6|…②求已知点 P8到直线 P5P6 的距离：过 P8 (x8, y8, z8) 点作直线 P5P6 的垂线,垂足为 Q,则P8到直线 P5P6的距离：|P8Q|P8Q⊥P5P6 P5P6的方向向量:∵|P5P6|=√[(x6-x5)2+ (y6-y5)2+(z6-z5)2]…③S= Sx i + Sy j + SZ k= ( x6-x5, y6-y5 , z6-z5 )/ √[(x6-x5)2+ (y6-y5)2+(z6-z5)2]∴Sx i= ( x6-x5 )/ √[(x6-x5)2+ (y6-y5)2+(z6-z5)2]iSy j= ( y6-y5 )/ √[(x6-x5)2+ (y6-y5)2+(z6-z5)2]jSZ k= ( z6-z5 )/√[(x6-x5)2+ (y6-y5)2+(z6-z5)2] k∴直线P5P6 的点向式方程：( x-x5,) / ( x6-x5) =( y-y5 ) /( y6-y5 ) =( z-z5 )/( z6-z5 )现令 Q点 对应于参数 tq , 则上述方程化为参数方程：x=x5+( x6-x5) tq ,y=y5+ ( y6-y5 ) tq,z=z5+ ( z6-z5 )tq∴Q=Q(x5+( x6-x5 ) tq ,y5+ ( y6-y5 ) tq,z5+ ( z6-z5 )tq)=(Qx, Qy, QZ) ∵ P8Q⊥P5P6∴P8Q·P5P6=0∴(Qx-x8)Sx+(Qy-y8)Sy+ (QZ-z8)SZ =0(Qx-x8) ( x6-x5)+(Qy-y8)( y6-y5 ) + (QZ-z8)( z6-z5 )=0即(x5+( x6-x5)t -x8) ( x6-x5)+(y5+ ( y6-y5 )t-y8)( y6-y5 ) + (z5+ ( z6-z5 )t-z8)( z6-z5 )=0 解得 tq =[(x8 -x5) ( x6-x5)+(y8-y5)( y6-y5 )+ (z8-z5)( z6-z5 )]/[( x6-x5)2+( y6-y5 )2+ ( z6-z5 )2]代入Q点 坐标的参数表示式从而得到交点 Q 的坐标为: Qx = x5+( x6-x5 ) tq=x5+( x6-x5 ) [(x8 -x5) ( x6-x5)+(y8-y5)( y6-y5 )+ (z8-z5)( z6-z5 )]/[( x6-x5)2+( y6-y5 )2+ ( z6-z5 )2]=Qy = y5+ ( y6-y5 ) tq=y5+ ( y6-y5 ) [(x8 -x5) ( x6-x5)+(y8-y5)( y6-y5 )+ (z8-z5)( z6-z5 )]/[( x6-x5)2+( y6-y5 )2+ ( z6-z5 )2]QZ = z5+ ( z6-z5 )tq=z5+ ( z6-z5 )[(x8 -x5) ( x6-x5)+(y8-y5)( y6-y5 )+ (z8-z5)( z6-z5 )]/[( x6-x5)2+( y6-y5 )2+ ( z6-z5 )2]同理D点的坐标为: Dx = x5+( x6-x5 ) td=x5+( x6-x5 ) [(x7-x5) (x6-x5)+(y7-y5)(y6-y5)+ (z7-z5)(z6-z5)]/[(x6-x5)2+(y6-y5)2+ (z6-z5)2]=/|P5P6|2Dy = y5+ ( y6-y5 ) td=y5+ ( y6-y5 ) [(x7-x5) (x6-x5)+(y7-y5)(y6-y5)+ (z7-z5)(z6-z5)]/[(x6-x5)2+(y6-y5)2+(z6-z5)2]DZ = z5+ ( z6-z5 )td=z5+ ( z6-z5 )[(x7-x5) (x6-x5)+(y7-y5)(y6-y5)+ (z7-z5)( z6-z5 )]/[(x6-x5)2+(y6-y5)2+ (z6-z5)2]且 DP7=(x7-x5-( x6-x5) td,y7-y5-( y6-y5) td,z7-z5-( z6-z5)td )其中td=[(x7-x5) (x6-x5)+(y7-y5)(y6-y5)+ (z7-z5)( z6-z5 )]/[(x6-x5)2+(y6-y5)2+ (z6-z5)2]DP7={[(z5x6-x5z6+z6x8-x6z8+z8x5-x8z5)(z6-z5)-(x5y6-y5x6+x6y8-y6x8+x8y5-y8x5)(y6-y5)]i+[(z5y6-y5z6+z6y8-y6z8+ z8y5-y8z5)(z6-z5)-(y5x6-x5y6+y6x8-x6y8+ y8x5-x8y5)(x6-x5)]j+[(x5z6-z5x6+x6z8-z6x8+ x8z5-z8x5)(x6-x5)+(y5z6-z5y6+y6z8-z6y8+ y8z5-z8y5)(y6-y5)]k}/((x6-x5)2+(y6-y5)2+(z6-z5)2)…④由 P8Q=(x5-x8+( x6-x5) tq ,y5-y8+ (y6-y5) tq ,z5-z8+ (z6-z5) tq )代入求得的 tq ,则有|P8Q|=√{[ (x5-x8)((x6-x5)2+(y6-y5)2+(z6-z5)2)+(x6-x5)((x8-x5) (x6-x5)+(y8-y5)(y6-y5)+ (z8-z5)( z6-z5 ))]2+[ ( y5-y8)((x6-x5)2+(y6-y5)2+(z6-z5)2)+(y6-y5)((x8-x5) (x6-x5)+(y8-y5)(y6-y5)+ (z8-z5)( z6-z5 ))]2+[ (z5-z8)((x6-x5)2+(y6-y5)2+(z6-z5)2)+(z6-z5)((x8-x5) (x6-x5)+(y8-y5)(y6-y5)+ (z8-z5)( z6-z5 ))]2}/((x6-x5)2+(y6-y5)2+(z6-z5)2)=√{[(x5y6-y5x6+x6y8-y6x8+ x8y5-y8x5)(y6-y5) - (z5x6-x5z6+z6x8-x6z8+ z8x5-x8z5)(z6-z5)]2+[(y5x6-x5y6+y6x8-x6y8+ y8x5-x8y5)(x6-x5)- (z5y6-y5z6+z6y8-y6z8+ z8y5-y8z5)(z6-z5)]2+[(z5x6-x5z6+z6x8-x6z8+ z8x5-x8z5)(x6-x5)+(z5y6-y5z6+z6y8-y6z8+ z8y5-y8z5)(y6-y5)]2}/((x6-x5)2+(y6-y5)2+(z6-z5)2)同理|P7D|=√{[(x5y6-y5x6+x6y7-y6x7+ x7y5-y7x5)(y6-y5) - (z5x6-x5z6+z6x7-x6z7+ z7x5-x7z5)(z6-z5)]2+[(y5x6-x5y6+y6x7-x6y7+ y7x5-x7y5)(x6-x5)- (z5y6-y5z6+z6y7-y6z7+ z7y5-y7z5)(z6-z5)]2+[(z5x6-x5z6+z6x7-x6z7+ z7x5-x7z5)(x6-x5)+(z5y6-y5z6+z6y7-y6z7+ z7y5-y7z5)(y6-y5)]2}/|P5P6|2=√{[(x5y6-y5x6+x6y7-y6x7+ x7y5-y7x5)(y6-y5) - (z5x6-x5z6+z6x7-x6z7+ z7x5-x7z5)(z6-z5)]2+[(y5x6-x5y6+y6x7-x6y7+ y7x5-x7y5)(x6-x5)- (z5y6-y5z6+z6y7-y6z7+ z7y5-y7z5)(z6-z5)]2+[(z5x6-x5z6+z6x7-x6z7+ z7x5-x7z5)(x6-x5)+(z5y6-y5z6+z6y7-y6z7+ z7y5-y7z5)(y6-y5)]2}/((x6-x5)2+(y6-y5)2+(z6-z5)2)∵N1| /|P5P6|=√[[(y6-y7 )(z5-z7)-(z6-z7)(y5-y7 )]2 +[ (x6-x7 )(z5-z7)- (z6-z7)(x5-x7 ) ] 2+[ (x6-x7 )(y5-y7 )-(y6-y7 )(x5-x7 )] 2]/√[(x6-x5)2+ (y6-y5)2+(z6-z5)2]|P7D|=|N1|/|P5P6|∴|P7D|=√[[(y6-y7 )(z5-z7)-(z6-z7)(y5-y7 )]2 +[ (x6-x7 )(z5-z7)- (z6-z7)(x5-x7 ) ] 2+[ (x6-x7 )(y5-y7 )-(y6-y7 )(x5-x7 )] 2]/√[(x6-x5)2+ (y6-y5)2+(z6-z5)2]…⑤同理可得:|P8Q|=√[[(y6-y8 )(z5-z8)-(z6-z8)(y5-y8)]2 +[ (x6-x8 )(z5-z8)- (z6-z8)(x5-x8)] 2+[ (x6-x8 )(y5-y8 )-(y6-y8)(x5-x8 )] 2]/[(x6-x5)2+ (y6-y5)2+(z6-z5)2]…⑥其实, DP7‖QP9 , |P8Q|=|P9Q| 且 QP9=DP7|P9Q|/|P7D|∴ P9=Q + QP9=Q +DP7 |P8Q| /|P7D| …⑦上式即为P9的一种向量坐标转换公式 P5P6的方向向量:2.jpg