13.5 Parametric Equations of Lines

13.5 Parametric Equations of Lines

A line is defined by a vector multiplied t times in 2-space.

You can multiply each component by t to find the vector’s parametric equations.

Take the vector <2,3>:

x would equal 2t and

y would equal 3t.

To translate a vector so that it doesn’t pass through the origin, we add another vector.

For example, if the vector must go through the point (1,0), we add the vector <1,0> to the original vector.

L= <1,0>+t<2,3>, where t is assumed to be defined as all real numbers

x=1+2t

y=0+3t

At time t=0, x and y equal 1 and 0.

This method can be easily applied in 3-space by including the third dimension.

The general formula for a 3-space vector that must pass through a specific point for V=<a,b,c> and point (x,y,z) is:

L=<x,y,z>+t<a,b,c>

The equation above is known as the vector equation of a line.

This can be simplified into parametric equations for each component by simply simplifying the equation for each point:

x=at+x

y=bt+y

z=ct+z

For example, a line that has a point at(<3,2,6) and is parallel to vector <1,2,5>

is expressed as

L = <3,2,6>+t<1,2,5>
x=3+t

y=2+2t

z=6+5t

To figure out if two lines are parallel, figure out if they are scalar multiples for each other. For instance, if one line was parallel to <1,2,3> and the other was parallel to <-2,-4,-6>, they would be parallel because all the terms in the first vector can be multiplied by the same coefficient (in this case it’s -2) to become all the terms in the second vector.

To find where two lines intersect (if they do), take the two equations for the x component of the parametric, set them equal to each other, and solve for t(1) in terms of t(2), or vice versa. If the lines intersect, these values of t will set the y and z variables also equal. If they do not come out as equal with the initially found t values, the lines do not intersect. We’ll use the lines

x=3+2t(1)

y=t(1)-1

z=3t(1)

and

x=2t(2)

y=5

z=t(2)+4

First we set them equal to each other:

2t(2)=3+2t(1)

5=t(1)-1

t(2)+4=3t(1)

The easiest t to find is t(1):

5=t(1)-1

t(1)=6

Plugging this into

t(2)+4=3t(1),

we find t(2)

t(2)+4= 18

so t(2)= 14

Finally, plugging those values into the third equation

2t(2)=3+2t(1),

we test for intersection

2(14)=3+2(6)

28 is not equal to 15

So the lines do not intersect.

If two lines are not parallel and do not intersect, they are skew.

Using time as a part of the equations of lines and vectors might be weird at first, but once you get used to it, it’s actually really easy and somewhat fun!

-Lance, Ben, and Jack <3 <3 <3