Contents [VG.1] [VG.2] [VG.3] [VG.4] [VG.5] [VG.6] [VG.7]

VG.5 Equations of Lines and Planes

In two-dimensions, a person needs a point and a slope to form a line on a graph. The point says where the line will exist and the slope says at what angle the line will run. A person could also put the information into a point slope equation to represent the line. It is the same concept in three dimensions, however a person needs more than just a slope to graph a line. In three dimensions, a person needs a point and a vector. Like a 2-D graph the point shows where the line will be at, but the vector is needed to show a slope in 3-D.

The Vector Equation for a Line

In this equation, r and t are variables, r $_{0}$ is where you plug the point in, and v is where you plug the vector in.

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The above equation can be expanded into three equations to describe the line in individual components. These equations are called parametric equations of the line.

Parametric Equations

In this equation, the values $x_{0},$ $y_{0},$ and $z_{0}$ are the individual components of the point, and the values a, b, and c are the individual components of the vector that is used to form the line, the values of the components of the vector are called direction numbers.

Another way of describing line L is to eliminate the parameter t from the parametric equations. This then forms the symmetric equations.

Symmetric Equations

MATH

Skew Lines - When two lines do not intersect and are not parallel

Planes

A plane is a little harder to describe than a line. With a line, a vector parallel to the line will sufficiently describe it. However, a vector parallel to a plane will not describe the plane. To describe a plane one needs to find a vector called the normal vector (n). A normal vector is a vector that is perpendicular to the plane.

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