Sec 13.5 Equations of Lines and Planes

1. Sec 13.5Equations of Lines and Planes A line L in three dimensional space is determined when we know a point on L and the direction of L.

2. Parametric and Symmetric Equations of the line L: Equating the components of the vector equation, we obtain the parametric equations: where t is a real number. Eliminating the parameter t from the parametric equations, we obtain the symmetric Equations of L: Note: The numbers a, b, and c are called the direction numbers of L.

3. Planes A plane Π in space is determined when we know a point in the plane and a vector perpendicular (orthogonal) to the plane.

4. The scalar equation of the plane Π Using the component forms of the vectors in the vector equation, we obtain the scalar equation: Note: The equation ax + by + cz + d = 0 is called the linear equation of the plane.

5. The distance from a point P to a plane Π A Distance Formula:

6. Sec 13.6Cylinders and Quadratic Surfaces A cylinder is a surface that consists of all lines (called rulings) that are parallel to a given line and pass through a given plane curve C. The curve C is called the generating curve of the cylinder. A quadric surface is the graph of a second degree equation in three variables, x, y, and z. By translation and rotation, these equations can be reduced to one of the two standard forms:

7. For graphs of quadric surfaces See Table 1 on page 844.