1. Section 3.5 Lines and Planes in 3-Space

2. Let n = (a, b, c) ≠ 0 be a vector normal (perpendicular) to the plane containing the point P 0 (x 0, y 0, z 0 ). For any point P(x,y, z) in the plane, the vector is orthogonal to n. That is, Hence, This is called the point-normal form of a plane. POINT-NORMAL EQUATION OF A PLANE

3. GENERAL FORM OF THE EQUATION OF A PLANE Theorem 3.5.1: If a, b, c, and d are constants and a, b, and c are not all zero, then the graph of the equation ax + by + cz + d = 0 is a plane having the vector n = (a, b, c) as a normal. This equation is called the general form of the equation of a plane.

4. GEOMETRIC INTERPRETATION OF A SYSTEM OF EQUATIONS Recall the solution to a system of linear equations with two variables and two equation corresponds to the intersection of two lines. Similarly, the solution of a system of three equations with three variables corresponds to the intersection of three planes.

5. VECTOR FORM OF EQUATION OF A PLANE Let P(x, y, z) be any point in a plane and let P 0 (x 0, y 0, z 0 ) be a specific point in the plane. Let r 0 be the vector from the origin to P 0 (x 0, y 0, z 0 ), r be the vector from the origin to P(x, y, z), and n = (a, b, c) be a vector normal to the plane. Then, so the general equation of the plane can be rewritten as n · (r − r 0 ) = 0. This is called the vector form of the equation of a plane.

6. Let v = (a, b, c) ≠ 0 be a vector parallel to the line l in 3- sapce, and line l contains the point P 0 (x 0, y 0, z 0 ). For any point P(x, y, z) on the line l, the vector is parallel to v. That is, for some scalar t, Hence, (x − x 0, y − y 0, z − z 0 ) = (ta, tb, tc). That is, x = x 0 + ta, y = y 0 + tb, z = z 0 + tc, −∞ < t < ∞. These equations are called parametric equations for the line l. LINES IN 3-SPACE

7. VECTOR FORM OF THE EQUATION OF A LINE Let r = (x, y, z) be the vector from the origin to the point P(x, y, z), let r 0 = (x 0, y 0, z 0 ) be the vector from the origin to the point P 0 (x 0, y 0, z 0 ), and let v = (a, b, c) be the vector parallel to the line. Then and the equation of the line can be written as r − r 0 = tv Taking into account the range of t-values, this can be written as r = r 0 + tv (−∞ < t < ∞) This is called the vector form of the equation of a line in 3- space

8. DISTANCE BETWEEN AND POINT AND A PLANE Theorem 3.5.2: The distance D between a point P 0 (x 0, y 0, z 0 ) and the plane ax + by + cz + d = 0 is