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   Vectors and Parametric Equations

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NORM PROKUP



Cornell University
PhD. in Mathematics

Norm was 4th at the 2004 USA Weightlifting Nationals! He still trains and
competes occasionally, despite his busy schedule.






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PARAMETRIZING A LINE SEGMENT - CONCEPT

Norm Prokup

NORM PROKUP



Cornell University
PhD. in Mathematics

Norm was 4th at the 2004 USA Weightlifting Nationals! He still trains and
competes occasionally, despite his busy schedule.



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 * Explanation
 * Transcript

There are some situations that are useful to learn to parameterize. We commonly
parameterize line segments, and require knowledge of the starting and ending
positions. In order to understand how to parameterize line segments, students
should understand the concept of parametric equations. The tools we use to
parameterize a line can be useful when understanding how to parameterize a
circle.

vector equation line segment parametric equations

I want to talk about how to get a parametric equation for a line segment. Now
let's start with a line segment that goes from point a to x1, y1 to point b x2,
y2. And now we're going to use a vector method to come up with these parametric
equations. First of all let's notice that ap and ab are both vectors that are
parallel. So ap equals t times ab some scalar t times ab and you know that
because ap is smaller than ab that this is going to be some kind of compression
that means that t is going to be between 0 and 1 as long as point p is between
these 2 guys. Now let's take a look at the components for ap and ab I've named
point p x and y I'm going to use this point to derive my equation, my equation
has to have x and y.
Let's say that these are numbers that I know these coordinates here and here, ap
has components x-x1 and y-y1 and ab has components x2-x1 and y2-y1. So I have a
scalar of multiplication here and I'm going to multiply the t through and I get
tx2w-tx1, ty2-ty1 and that equals x-x1, y-y1, now to go from a vector equation
to parametric equations all I have to do is separate these by components. So
x-x1=tx2-tx1, tx2-tx1 let me add the x1 to both sides and I get x equals and I'm
going to pull this guy in front, x1-tx1 and then plus tx2 and that gives me x
equals, notice that I have a common factor of x1 I can pull out the 1-t and I'm
left with x1, 1-t times x1+t times x2.
Now because t is between 0 and 1 this is like a weighted average of x1 and x2
and now the ... same thing is going to happen for y, y-y1 equals ty sub 2 minus
ty sub 1 again I'll move the y1 term in front and I'll add y1 to both sides I'll
get y1-ty1+ty2 so this is after I factor the y1 out, I get 1-t, y1+ty2=y. So
these are my 2 equations right here and then putting them together x=1-tx1+tx2,
y=1-ty1+ty2 and these equations work for t again between 0 and 1. So all you
need to come up with a parametric equations for a line segment are the
coordinates of the 2 end points x1 y1 and x2 y2 and you can always use this
parameterization to get you that line segment.
more

Precalculus Vectors and Parametric Equations


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