Assignment 10:

Investigating Parametric Equations and Magnitude of Line Segments

by

Ángel M. Carreras Jusino

Goals:

1. Write parametric equations of a line segment through (7, 5) with slope of 3.

2. Graph the line segment using your equations.

3. As a line segment, it will have end points. Explore how you would chose endpoints of such that the two distances from (7, 5) are 2 units and 3 units.

Write parametric equations of a line segment through (7, 5) with slope of 3.There are various ways to write parametric equations of a line segment given a point in it and its slope. One way is to use the coordinates of the point given as the constant terms in the parametric equations. In the case that we are exploring 7 would be the constant for

x(t) and 5 the constant fory(t). Now, with the given slope we know howx(t) varies with respect toy(t). For a slope of 3, for each unit thatxcoordinate travels theycoordinate travels 3 units, this values serve as coefficients of the linear term of the parametric equations. So the parametric equations for the case given is given by

Note that these parametric equations initial point (

t= 0) is the point given (7, 5).

Graph the line segment using your equation.

Explore how you would chose endpoints of the line segment that the distance to the endpoints from (7, 5) are 2 units and 3 units.We want to find the value of

tsuch that the distance from (x(t),y(t)) to (7, 5) is the one required.Let

rbe the distance required, using the distance formula we have

For the first endpoint we want it to be 2 units from (7, 5), therefore using the formula above with

r= 2 we get thetvalue

Now, using this value in the parametric equations would give us an endpoint in the segment line 2 units from (7, 5), but also using the additive inverse of this value of

twould give us an endpoint at the same distance from (7, 5) but in the opposite direction. For this case we will use the additive inverse oftand for the next case (r= 3) we will use the value that the formula provide.Replacing the value of

tfound in the parametric equations we get

For the other endpoint we want it to be 3 units from (7, 5), therefore using again the formula with

r= 3 we get thetvalue

Replacing the value of

tfound in the parametric equations we get

In this case, if we want to create a segment that lies on the line described by the given parametric equations with a particular length

r, we have to choseaandbvalues such that |a-b| =r, then using the values

in the parametric equations will get us a segment with the desired length.

This can be generalized for any given parametric equations of a line segment in the following way:

- Let
vbe the magnitude of of the segment defined byt= 0 andt= 1 (or any two values oftthat are which their difference is 1).- Let
rbe the desired length of the segment.- Chose
aandbvalues such that |a-b| =r.- Using the values of

in the given parametric equations will get us the segment with the desired length.