Planes in the spaceAn equation of a plane in space can be obtained from a point in the plane and a vector normal (perpendicular) to the plane. Consider the plane containing the point P(x_{1},y_{1},z_{1}) having a nonzero normal vector n=<a,b,c>, as shown in the next figure: This plane consists of all points Q(x,y,z) for which vector is orthogonal to n. Using the dot product, you can write the following: n·=0 <a, b, c>·<xx_{1}, yy_{1}, zz_{1}>=0 a(xx_{1}) + b(yy_{1}) + c(zz_{1}) = 0 The third equation of the plane is said to be in standard form.STANDARD EQUATION OF A PLANE IN SPACE. The plane containing the point (x_{1},y_{1},z_{1}) and having normal vector n = <a,b,c> can be represented by the standard form of the equation of a plane:
By regruping terms, you obtain the general form of the equation of a plane in space:
Given the general form of the equation of a plane, it is easy to find a normal vector to the plane. Simply use the coefficients of x, y and z and write n = <a,b,c> Find the general equation of the plane containing the points (2,1,1), (0,4,1), and (2,1,4) Solution: The component forms of u and v are: u = <0  2, 4  1, 1  1> = <2, 3, 0> v = <2  2, 1  1, 4  1> = <4, 0, 3> and it follows that n = u x v =
=9i + 6j +12k = <a,b,c> is normal to the given plane. Using the direction numbers for n and the point (x_{1},y_{1},z_{1})=(2,1,1), you can determine an equation of the plane to be
PARAMETRIC EQUATIONS OF PLANES IN SPACE. Supose that u = <u_{1},u_{2},u_{3}> and v = <v_{1},v_{2},v_{3}> are nonparallel vectors that lie in a plane V. Let P_{0}=(x_{0},y_{0}.z_{0}) be any fixed point that lies on V. We can think of P_{0} as an "origin" in the plane. Any point P=(x,y,z) on the plane can be reached from P_{0} by adding a scalar multiple of u and then a scalar multiple of v, as illustrate in the following picture: The position vector can be obtained by vector addition as: = + u + v If we write out this vector equation in coordinates, then we get three scalar parametric equations for the plane:
The parameters and may be thought of as coordinates in the plane V: Each point of V is determined by specifying the values of and .
