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Boundless Calculus
Advanced Topics in Single-Variable Calculus and an Introduction to Multivariable Calculus
Vectors and the Geometry of Space
Calculus Textbooks Boundless Calculus Advanced Topics in Single-Variable Calculus and an Introduction to Multivariable Calculus Vectors and the Geometry of Space
Calculus Textbooks Boundless Calculus Advanced Topics in Single-Variable Calculus and an Introduction to Multivariable Calculus
Calculus Textbooks Boundless Calculus
Calculus Textbooks
Calculus
Concept Version 10
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Equations of Lines and Planes

A line is a vector which connects two points on a plane and the direction and magnitude of a line determine the plane on which it lies.

Learning Objective

  • Explain the relationship between the lines and three dimensional geometric objects


Key Points

    • The slope of the line, and the plane it lies on, is the angle of inclination of that line.
    • A line is a two dimensional representation of a three dimensional geometric object, a plane.
    • The parametric equation of a line is written as: <x,y,z> = <x0+at,y0+bt,z0+ct><x, y, z> \ = \ <x_0 + at, y_0+bt, z_0 + ct><x,y,z> = <x​0​​+at,y​0​​+bt,z​0​​+ct>

Terms

  • slope

    also called gradient; slope or gradient of a line describes its steepness

  • vector

    a directed quantity, one with both magnitude and direction; the signed difference between two points


Full Text

Lines and Planes

A line is described by a point on the line and its angle of inclination, or slope. Every line lies in a plane which is determined by both the direction and slope of the line. A line is essentially a representation of a cross section of a plane, or a two dimensional version of a plane which is a three dimensional object.

Equations of Lines and Planes

The components of equations of lines and planes are as follows:

A line in three dimensional space is given by a point, P0(xo,yo,zo)P_0 (x_o,y_o,z_o)P​0​​(x​o​​,y​o​​,z​o​​), or a plane, MMM, and its direction. This direction is described by a vector, v\mathbf{v}v, which is parallel to plane and PPP is the arbitrary point on plane MMM.

The position vector of point P0P_0P​0​​ is called r0\mathbf{r}_0r​0​​ and the position vector of point PPP is called r\mathbf{r}r. The vector from PPP to P0P_0P​0​​ is called vector a\mathbf{a}a. Vectors a\mathbf{a}a and v\mathbf{v}v are parallel to each other.

Now, we can use all this information to form the equation of a line on plane MMM. 

The vector equation of a line is:

r=r0+tv\mathbf{r} = \mathbf{r}_0 + t \mathbf{v}r=r​0​​+tv

where ttt represents the location of vector r\mathbf{r}r on plane MMM.

The parametric equation of a line can be written as: 

$x = x_0 + at \\ y=y_0 +bt \\ z = z_0 +ct$

Or the more compact form:

 <x,y,z> = <x0+at,y0+bt,z0+ct><x, y, z> \ = \ <x_0 + at, y_0+bt, z_0 + ct><x,y,z> = <x​0​​+at,y​0​​+bt,z​0​​+ct>

Vertical Line, Graphed

Vertical line x=ax = ax=a, lying on the xyxyxy-plane (z=0z=0z=0).

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