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3.7 Planar Equations 高等代数与解析几何 3.7 平面方程 3.7.1 Planar equations decided by a point in plane and azimuthal vector Given a point M 0 and two vectors a , b that are not collinear in space, then the plane via this point that parallel with vectors a, b is uniquely confirmed, vectors a , b is called azimuthal vector of the plane. Take affine coordinate system {O; e1 , e2 , e3} in space, and let vector radius of point M 0 is OM 0 r0，the vector radius of an arbitrary point M in plan is OM r . By coplanar condition, MoM u a v b can be written as r r0 u a v b . 2 高等代数与解析几何 3.7 平面方程 The planar equation above is called vector form parametric equation of plane . Let the coordinates of points M 0 , M are ( x0 , y0 , z0 ),( x, y, z), respectively, then r0 ( x0 , y0 , z0 ), r ( x, y, z ) . Let a ( X 1 , Y1 , Z1 ), b ( X 2 , Y2 , Z 2 ) then x x0 X 1u X 2 v y y0 Y1u Y2 v z z0 Z1u Z 2 v 3 高等代数与解析几何 3.7 平面方程 The equations above are called coordinate form arametric equations of plan , where u, v are parameters. Point place form equation of plane. x x0 y y0 z z0 X1 Y1 Z1 0 X2 Y2 Z2 Example 3.32 There are three points M1 ( x1 , y1 , z1 ), M 2 ( x2 , y2 , z2 ), M 3 ( x3 , y3 , z3 ) that are not collinear, find the equation of plane via points M1, M 2 , M 3. 4 高等代数与解析几何 3.7 平面方程 Solution By vector form parametric equation, remove parameters u, v, respectively, we obtain ( r r1 , r2 r1 , r3 r1 ) 0. x x1 y y1 z z1 x2 x1 y2 y1 z2 z1 0 x3 x1 y3 y1 z3 z1 can be rewritten as x y z 1 x1 y1 z1 1 0 x2 y2 z2 1 x3 y3 z3 1 5 高等代数与解析几何 3.7 平面方程 The above two equations are both called three points form equation of plane. Specially, if three known points are intersection points of plane and three axes M 1 (a, 0, 0), M 2 (0, b, 0), M 3 (0, 0, c)(abc 0), then we obtain intercept form equation of plane x y z 1. a b c 6 高等代数与解析几何 3.7 平面方程 3.7.2 General equation of plane Since any plane can be confirmed by a point M 0 ( x0 , y0 , z0 ) in it and its azimuthal vector a ( X 1 , Y1 , Z1 ), b ( X 2 , Y2 , Z 2 ) by point , direct form it is written as Ax By Cz D 0 (*) where Y1 Z1 Z1 X1 X1 Y A ,B ,C Y2 Z2 Z2 X2 X 2 Y2 7 高等代数与解析几何 3.7 平面方程 Equation（*）is called general form equation of a plane. Theorem 3.4 Any planar equation in space can be represent- ed as a linear equation on variables x, y, z , Conversely, linear equation on each every variable x, y, z represents a plane. Let’s discuss several special cases of equation（*），i.e. while some coefficients or constants of it equal to zeroes, plane have some special positions for coordinate system. 8 高等代数与解析几何 3.7 平面方程 1. If D 0 , (*) is Ax By Cz 0 origin (0, 0, 0)satisfies , the equation at this time, hence plane is via the origin. Conversely, if the plane is via the origin, then D 0 obviously. 2. One of A, B, C is zero. On the condition of D 0, if A 0 , plane is parallel with axis x , if B 0, the plane is parallel with axis y . On the condition of C 0 , the plane is parallel with axis z . 3. Two of A, B, C are zeroes, We have following conclusions. Plane is parallel with coordinate plane yoz if and only if D 0, B C 0.The plane is just coordinate plane yoz if and only if 9 高等代数与解析几何 3.7 平面方程 D 0, B C 0. Similarly for other cases. 3.7.3 Normal equation of plane Given a fixed point M 0 and a vector n that is not equal to zero, then a plane via the point M 0 is uniquely confirmed, which is vertical with the vector n , and the vector is called normal vector of the plane. Under rectangular coordinates system in space {o; i, j, k} , let vector radius of point M 0 be OM 0 r0 , and the vector radius of an arbitrary point M in plane be OM r . The point M 10 高等代数与解析几何 3.7 平面方程 is in the plane if and only if the vector M 0 M r r0 is vertical with the vector n , i.e. A( x x0 ) B( y y0 ) C ( z z0 ) 0 Let D ( Ax0 By0 Cz0 ), then the equation above is Ax By Cz D 0 which is called point normal form equation of plane. Take the origin O as a point M 0 in plan specially, foot of a perpendicular line that constructed toward plane is p ，the 0 normal vector of is identity vector n . If the plane is not via 11 高等代数与解析几何 3.7 平面方程 0 the origin, the positive direction of n and the vector op have the same direction, while the plane is via the origin, take either of the two directions that is perpendicular with the plane as the 0 positive direction of n . Let OP p , then the normal equation of vector is 0 n r p 0 0 Let r ( x, y, z), n (cos 1,cos 2 ,cos 3 ) ，then x cos 1 y cos 2 z cos 3 p 0 which is called coordinate form normal equation of plane. 12 高等代数与解析几何 3.7 平面方程 Two characters of normal equation of plane. 1. The coefficient of linear factor is component of identity vector, whose sum of squares is equal to 1； 2. The constant is not more than zero. According to the two characters of normal equation of plane, we can easily transform the general form equation of plane into following form n r D 0. Multiply both side of the above equation by 1 1 k |n| A2 B 2 C 2 13 高等代数与解析几何 3.7 平面方程 we can obtain normal equation of plane, hence k is called normal factor. Example 3.33 Given two points M 1 (1, 2,3), M 2 (3, 0 1)， find the equation of perpendicular bisector plane of line segment M 1M 2 . Answer x y 2z 1 0 14 The End of Section 3.7