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Chapter 9 – Relationships between points, lines and planes. By: Narinder Lall, Danish Khan, Ziyad Ishmael. 9.1 – Intersection of two lines and a line and a plane. Intersection between lines and planes can occur in 3 ways: Case 1: The line L intersects the plane π at a point.
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Chapter 9 – Relationships between points, lines and planes By: Narinder Lall, Danish Khan, Ziyad Ishmael
9.1 – Intersection of two lines and a line and a plane • Intersection between lines and planes can occur in 3 ways: • Case 1: The line L intersects the plane π at a point. • Case 2: The line L does not intersect the plane and is parallel to the plane. This case has no solutions. • Case 3: The line L lies on the plane π. This case has infinite solutions. • Intersection between two lines can occur in 4 ways: • Case 1: The lines intersect at a point. • Case 2: The lines are parallel and therefore it has no intersection point, so no solution. • Case 3: Two lines are not parallel and do not intersect, giving no solution. These lines are called skew lines. • Case 4: The lines are parallel and coincident, therefore they have infinite solutions.
9.2 – Systems of equations • Elementary row operations: • Multiple an equation by a non-zero constant. • Interchange any pair of equations. • Add a multiple of one equation to the second to replace the second equation. • There are two methods to solve for system of equations: • Using variables • Using matrices
9.3 – Intersection of two planes • Finding the solution for three unknowns of two equations of planes in Cartesian form. • Two planes can intersect in 3 different ways: • Case 1: Two planes intersect along a line and will have infinite number of P.O.I • Case 2: Two planes are parallel and non-coincident. This will have no P.O.I • Case 3: Two planes can be coincident and will have an infinite number of P.O.I • For two given planes, if n1 = kn2, k for some scalar then two given planes either are coincident or parallel and non-coincident. • For two given planes, if n1 = kn2, k for some scalar then two given planes intersect along a line.
9.4 – The intersection of three planes • Intersection of three planes has two types of systems: • Consistent systems: system that contains solution(s) • Case 1: There is one P.O.I therefore one solution. • Case 2a: The three planes intersect along a line and none of the planes are parallel, giving infinite solutions, it uses one parameter. • Case 2b: The three planes intersect along a line and π1 & π2 are both parallel and coincident, giving infinite solutions, it uses one parameter. • Case 3: The three planes intersect along a plane as all the three planes are parallel and coincident to each other, it uses two parameters and gives infinite solutions.
9.4 - continued • Inconsistent systems: system that contains no solutions. • Case 1: Three planes form a triangular prism. There is no common P.O.I for all three planes. • Case 2: Two non-coincident parallel planes each intersect a third plane. • Case 3: The three planes are parallel and non-coincident, this will have no P.O.I • Case 4: Two planes are coincident and parallel to the third plane, there is no common P.O.I for all three planes.
9.5 – Distance from a point to a line in R2 and R3 In R2, distance from point P0(x0,y0) to the line with equation Ax+By+C = 0 is In R3, the formula for the distance d from point P to the line r = r0 + sm, s Є R, is d = l m X QP l l m l
9.6- Distance from a point to a plane • Distance from a point P0(x0,y0,z0) to the plane with the equation Ax + By + Cz +D = 0 is d = • Distance between the skew lines can be found by 2 methods: • Method 1: Construct two parallel planes that are same distance apart as the skew lines, determine the distance between the two planes. • Method 2: 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.
