Publish on 18th November 2019
Category: Birds
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Parametric Equations of lines

Math 200Week 2 - Wednesday

Math 200

Main Questions for Today

How do we describe lines in space?How do we determine if two lines areparallel,intersecting, orskew?How do we figure out where lines intersection the coordinate planes?

Math 200

Describing the points on a particular line

A line is uniquely defined bytwo points(just like in 2D)We can draw a bunch of vectors between these points and the originWe can use these vectors to come up with a set of equations for any point on the line…

P1(x1,y1,z1)

P2(x2,y2,z2)

<x1,y1,z1>

<x2,y2,z2>

<a, b, c>

We now know that this vector is <x2-x1, y2-y1, z2-z1>, but we’ll just write <a,b,c> for short

Math 200

We want to describe anarbitrarypoint on the line in terms of the vectors we drew<x1, y1, z1> will get us to P1<a, b, c> gets us from P1to P2Scalar multiplesof <a, b, c> will go back and forth along the line!

(x,y,z)

<x1,y1,z1>

<a, b, c>

<x,y,z>

t<a, b, c>

Math 200

What’s the relationship between the vector that points directly to <x,y,z> and the other vectors?<x,y,z> = <x1,y1,z1> + t<a,b,c><x,y,z> = <x1+at, y1+bt, z1+ct>

(x,y,z)

<x1,y1,z1>

<a, b, c>

<x,y,z>

t<a, b, c>

Math 200

Example 1

Compute a set of parametric equations for the line containing the points A(1,3,4) and B(-2,1,7)First, wefind a vector parallel to the linev= <-2 - 1, 1 - 3, 7 - 4> = <-3, -2, 3>Now we pick a starting pointEither A or B will do

Math 200

You try one

Find a set of parametric equations for the line containing the points A(4,-2,0) and B(3,3,-1).Find another point on the line.

To get another point on the line, we can just plug in another “time” value fort.E.g. Whent= -1 we get (5, -7, 1)

Math 200

Another Example

Find a set of parametric equations for the line that passes through the point A(1,4,2) and is parallel to thexy-plane and theyz-plane.If it’s parallel to both thexy-plane and theyz-plane, then its direction vector must havex-andz-components that are zeroe.g. <0,1,0> would work

4

Math 200

Parallel, Intersecting, or skew

ParallelParallel lines have parallel direction vectorse.g.

The first line is parallel to the vector <-2,1,1>The second line is parallel to <4,-2,-2>Since, -2<-2,1,1> = <4,-2-2>, the lines are parallel.

Math 200

Skew vs intersectingSuppose we have two lines that are not parallele.g.

We know they’re not parallel because <-1,3,-2> is not parallel to <4,-1,1>.Theymayintersect, but they may pass right by one another…

Even if they do intersect, they may pass through their common point for differentt-values!

Rather than setting the equations equal as given, we’ll needdifferent lettersfor the parameter in each!

Pick two equations and solve foraandb

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