12.5 planes in 3-space A vector n that us perpendicular to a plane is called a normal vector to the plane. This vector determines how the plane is oriented as well As a family of parallel planes that all have n as a Normal vector to specify a particular plane, we select a point on the plane. Thus, a normal vector n to a plane and a point on the plane completely determine the plane This equation of the plane is equivalent to each of the following Geometric description of a plane versions: The plane P through p(0) = (x(0), y(0), z(0)) with normal vector n = consists of the tips of all vectors based At p(0) that are perpendicular to n n ha; b; ci ¢ hx ¡ x0; y ¡ y0; z ¡ z0i = 0 a (x ¡ x0) + b (y ¡ y0) + c (z ¡ z0) = 0 ax + by + cz = ax0 + by0 + cz0 n ¢ hx; y; zi = ax0 + by0 + cz0 Note that the equation ax + by + cz = d of a plane in 3-space is the direct generalization of the equation ax+by = c of a line in 2-space Example: xo.yo.to Find an equation of the plane through p(0) = (3, 1, 0) with a normal vector n = <3,2,-5> 3 (x ¡ 3) + 2 (y ¡ 1) ¡ 5z = 0 3x ¡ 9 + 2y ¡ z ¡ 5z = 0 3x + 2y ¡ 5z = 11 Equations of a plane Vector form: n ¢ hx; y; zi = d Scalar form: a (x ¡ x0) + b (y ¡ y0) + c (z ¡ z0) = 0 Note that it n is normal to a plane p then so is every nonzero scalar multiple. x + y + z = 1; 4x + 4y + 4z = 4 On the other hand, two planes P and p' are parallel if they have a common normal vector x + y + z = 1; x + y + z = 2 In general, a family of parallel planes is obtained by choosing a normal vector n = and varying the constant D in the equation ax + by + cz = d ax + by + cz = d Points that lie on a line are called collinear. If we are given three points P, q, and r that are not collinear, then there is just one plane passing through P, q, and r A plane p also can be determined by any of the following: Two lines that intersect in a single point A line and a point not on the line Two distinct parallel lines The intersection of a plane p with a coordinate plane or a plane parallel to a coordinate plane is called a trace. The trace is a one unless p is parallel to the coordinate plane (in which case, the trace is empty or is P itself) In most cases, you can picture a plane by determining where it intersects the coordinate axes. However, if the plane is parallel to one of the axes and does not intersect it, or the plane passes through the origin, and therefore intersects all three axes there, we can use the normal vector to determine how the plane is oriented