**Calculus and Vectors: Chapter 9**

**Contents**

Section 9.1 - The Intersection of a Line with a Plane and the Intersection of Two Lines

**9.1**

Section 9.2 - Systems of Equations

Section 9.3 - The Intersection of Two Planes

Section 9.4 - The Intersection of Three Planes

Section 9.5 - The distance form a Point to a Line in R^2 and R^3

Section 9.6 - The Distance from a Point to a Plane

**9.2**

**9.3**

**9.4**

**9.5**

**9.6**

Intersection between a line and plane

case 1

case 2

case 3

Case 1: The line L intersects the plane at exactly one point, P.

Case 2: The line L does not intersect the plane so it is parallel to the plane. There are no points of intersection.

Case 3: The line L lies on the plane Every point on L intersects the plane. There are an infinite number of points of intersection.

Intersection between two lines

Intersecting lines

case 1

Non-Intersecting lines

case 2

case 3

case 4

Case 1: The lines are not parallel and intersect at a single point.

Case 2: The lines are coincident, meaning that the two given lines are identical. There are an infinite number of points of intersection.

Case 3: The two lines are parallel, and there is no intersection.

Case 4: The two lines are not parallel, and there is no intersection. The lines in this case are called skew lines. (only exist in R3)

Example

Number of Solutions to a Linear System of Equations

Key Ideas

Definition of Equivalent Systems

A linear system of equations can have zero, one, or an infinite number of solutions.

Two systems of equations are defined as equivalent if every solution to one system is also a solution to the second system of equations, and vice versa.

Elementary Operations Used to Create Equivalent Systems

1. Multiply an equation by a nonzero constant.

2. Interchange any pair of equations.

3. Add a nonzero multiple of one equation to a second equation to replace the second equation.

Consistent and Inconsistent Systems of Equations

A system of equations is consistent if it has either one solution or an infinite number of solutions. A system is inconsistent if it has no solutions.

Key Ideas cont.

consider when finding the solutions to a system of two equations in two unknowns

Case 1: These equations represent two parallel and non-coincident lines.

Case 2: These equations represent two parallel and coincident lines.

Case 3: These two equations represent two intersecting, non-coincident lines.

Example

Solve the following systems using elementary operations:

Possible Intersections for Two Planes

Case 1: Two planes can intersect along a line. The corresponding system of equations will therefore have an infinite number of solutions.

Case 2: Two planes can be parallel and non-coincident. The corresponding system of equations will have no solutions.

Case 3: Two planes can be coincident and will have an infinite number of solutions.

Key Ideas

Solutions for a System of Equations Representing Two Planes

The system of equations corresponding to the intersection of two planes will have either zero solutions or an infinite number of solutions.

It is not possible for two planes to intersect at a single point

Intersection of Two Planes and their Normals

If the planes π1 and π2 have n1 and n2 as their respective normals, we know the following:

1. If n1 = kn2 for some scalar, k, the planes are coincident or they are parallel and non-coincident. If they are coincident, there are an infinite number of points of intersection. If they are parallel and non-coincident, there are no points of intersection.

2. If n1 does not equal kn2, the two planes intersect in a line. This results in an infinite number of points of intersection.

Example

Possible Intersections for Three Planes

There is just one solution to the

corresponding system of equations.

There are an infinite number of

solutions to the related system of equations.

The three planes intersect along a line and are mutually non-coincident.

There are an infinite number of

solutions to the related system of equations.

Two planes are coincident, and the third plane cuts through these t wo planes intersecting along a line.

Three planes are coincident, and there are an infinite number of solutions to the related system of equations.

Inconsistent systems of equations that represent three planes

Three planes form a triangular prism. Also the normals of the three planes are not scalar multiples of each other, so there are no solutions.

In this case, two parallel planes, each intersecting a third plane. Each of the parallel planes has a line of intersection with the third plane, but there is no intersection between all three planes so there are no solutions.

All three planes are parallel so there are no solutions.

In this case two planes are coincident and the third plane is parallel. There are no solutions.

Example

Distance Between a Point and a Line in R2

Use the formula on the right when trying to determine the perpendicular distance, which is also always the shortest distance, from a point (x0,y0) to a line represented by the formula: Ax+By+C=0

Distance Between a Point and a Line in R3

The most efficient way to find the distance bet ween a point and a line in R3 is to use the cross product.

Use the formula on the right when you are calculating the perpendicular distance from a point P to a line represented by the formula: r = r0+ sm , sER

Example

Distance Between a Point and a Plane

Use the formula on the right when trying to de termine the perpendicular distance, which is also always the shortest distance, from a point (x0,y0,z0) to a plane represented by the formula: Ax+By+Cz+D=0

Distance Between Skew Lines

Method 1

To determine the distance

between the given skew lines, two parallel planes are constructed that are the same distance apart as the skew lines. Determine the distance bet ween the two planes

Method 2

To determine the coordinate s 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

Example

work cited

Kirkpatrick, Chris, Peter W. D. Crippin, and Robert Donato. <i>Calculus and Vectors</i>. Toronto, Ont.: Nelson Education, 2009. Print.