Lines and Planes in Space 11.5 Lines and Planes in Space

Lines in Space To determine a line L, we need a point P(x1,y1,z1) on L and a direction vector for the line L. The parametric equations of a line in space is If a, b and c are nonzero, we can eliminate parameter to obtain the following symmetric equations:

Examples Find sets of parametric equations and symmetric equations of the line 1) through two points 2) through the point (2,3,1) and is parallel to

Equation of a Plane To determine a plane, we need a point P(x1,y1,z1) in the plane and a normal vector that is perpendicular to the plane. The standard equations of a plane in space is By regrouping terms, we obtain the general form of the line:

Examples Find an equation of the plane passing 1) through the points 2) through the points (3,2,1), (3,1,-5) and is perpendicular to

Planes in Space Two planes in space with normal vectors n1 and n2 are either parallel or intersect in a line. They are parallel if and only if their normal vectors are. They are perpendicular if and only if their normal vectors are. The angle between two planes is equal to the angle between the normal vectors are given by To find the line of intersection between two planes, solve the system of two equations with three unknowns. The line of intersection is parallel to

Examples Given two planes with equations 1) Find the angle between two planes. 2) Find the line of intersection of the planes.

Distance The distance between a plane (with normal vector n) and a point Q (not in the plane) is where P is any point in the plane. The distance between a line (with direction vector v) in space and a point Q is where P is any point on the line.

Examples Given two planes with equations 1) Find the distance between the point (1,1,0) and the plane P1. 2) Find the distance between the point (1,-2,4) and the line 3) Show that P1 and P2 are parallel. 4) Find the distance between P1 and P2 .

Distance Formulas The distance between a plane with equation and a point Q(x0,y0,z0) (not in the plane) is The distance between a line in the plane and a point Q(x0,y0) (not on the line) is