Equations for lines

Lines in the space

In the plane, slope is used to determine an equation of a line. In space it is more convenient to use vectors to determine the equation of a line.

In the following figure, consider the line L through the point P(x₁,y₁,z₁) and parallel to the vector
v=<a,b,c>. The vector v is a direction vector for the line L, and a, b, and c are direction numbers.

One way of describing the line L is to say that it consists of all points Q(x,y,z) for which the vector $\vec{P}$ is parallel to v. This mean that $\vec{PQ}$ is a scalar multiple of v, and you can write $\vec{PQ}$ =vt, where t is a scalar (a real number).

$\vec{PQ}$ = <x-x₁, y-y₁, z-z₁> = <at,bt,ct> = tv

Vector Equation of a Line in Space

A line L parallel to the vector v = <a,b,c> and passing through the point P(x₁,y₁,z₁) is represented by the vector equation:

(x,y,z)=(x₁,y₁,z₁)+t(a,b,c)

By equating corresponding components, you can obtain parametric equations of a line in the space.

Parametric Equations of a Line in Space

A line L parallel to the vector v = <a,b,c> and passing through the point P(x₁,y₁,z₁) is represented by the parametric equations:

x = x₁+at

y = y₁+bt

z = z₁+ct

If the direction numbers a, b, and c are all nonzero, you can eliminate the parameter t to obtain symmetric equations of the line.

Symmetric Equations of a Line in Space

A line L parallel to the vector v = <a,b,c> (where a, b, and c are all nonzero) and passing through the point P(x₁,y₁,z₁) is represented by the symmetric equations:

$\frac{x-x_1}{a}=\frac{y-y_1}{b}=\frac{z-z_1}{c}$

Find parametric and symmetric equations of the line L that passes through the point (1,-2,4) and is parallel to v = <2,4,-4>

Solution:
To find a set of parametric equations of the line, use the coordinates

x₁=1, y₁=-2, and z₁=4

and direction numbers: a=2, b=4, and c=-4.

You will obtain:

x=1+2t
y=-2+4t
z=4-4t

Parametric equations

Because a, b, and c are all nonzero, a set of symetric equiations is:

$\frac{x-1}{2}=\frac{y+2}{4}=\frac{z-4}{-4}$

Symmetric equations

Parametric Equations of a Line Through Two Points

Find a set of parametric equations of the line that passes through the points (-2,1,0) and (1,3,5)

Solution:
Begin by using the points P(-2,1,0) and Q(1,3,5) to find a direction vectro for the line passing through P and Q, given by:

v = $\vec{PQ}$ = <1-(-2), 3-1, 5-0> = <3,2,5> = <a,b,c>

Using the direction numbers a=3, b=2, and c=5 with the point P(-2,1,0), you can obtain the parametric equations:

x=-2+3t, y=1+2t and z=5t

Note:

As t varies over all real numbers, the parametric equations determine the points (x,y,z) on the line. In particular, note that t=0 and t=1 give the original points (-2,1,0) and (1,3,5)