Chapter 9 Review

1. Chapter 9 Review KulvirMankaranNavPinder

2. 9.1 • Line and Plane intersections can occur in three different ways • Line intersects the plane at exactly one point, one solution • Line is parallel to the plane, no solutions • Line lies on the plane, infinite solutions • Line and Line can Occur in four ways • Line intersects at a single point, one solution • Line intersects are parallel, no solution • Two lines are not parallel and don’t intersect, no solution (R3) (skew lines) • Two lines are coincident, infinite solution

3. 9.2 • Two linear equations can have zero, one or an infinite number of solutions (same as 9.1) • Elementary operations can be used to solve a system of equations • Multiply an equation by a nonzero constant • Interchange any pair of equations • Add a multiple of one equation to a second equation to replace the second equation • Consistent if solution is one or infinite • Inconsistent if it has no solutions

4. 9.3 • Two equations with 3 unknowns can have three relations • Two planes intersecting along a line, infinite solutions • Two planes can be parallel and non-coincident, no solutions • Two planes can be coincident, infinite solutions

5. 9.4 • Consistent Systems for three equations representing three planes • One solution, there is a single point • Infinite solutions, the solution uses one parameter • Three planes intersect along a line • Two planes are coincident, and the third plane cuts through these two planes • Infinite Solutions, uses two parameters. Three planes are coincident

6. Cont.. • Inconsistent Systems for three equations representing (no solutions) • Three planes form a triangular prism • Two non-coincident parallel planes intersect a third plane • The three planes are parallel and non-coincident • Two planes are coincident and parallel to the third plane

7. 9.5 • In R2, the distance from point P1(x1,y1) to the line with equation Ax + By + C = 0 is d=|Ax1 +By1 + C|/ √(A^2 + B^2) • In R3, the formula for the distance d from point P to the line r= r1 +sm, SER, is d=|m x QP|/|m|

8. 9.6 • The distance from a point p1(x1,y1,z1) to the plane with equation Ax + By+ Cz + D = 0 is d=|Ax1 + By1 + C1 +D|/√(A^2 + B^2 + C^2) • Distance between skew lines, two methods • Distance between two skew lines. Two parallel planes are constructed that are the same distance apart as the skew lines. Determine the distance between the two planes. • To determine the coordinates of the points that produce the minimal distance, use the fact that the general vector found by joining the two points is perpendicular to the direction vector of each line