Determining coordinates of a point on a line perpendicular to a vector

[adrimare]

Homework Statement

The two lines L1: r = (-1,1,0)+ s(2,1,-1) and L2: r = (2,1,2) + t(2,1,-1) are parallel but do not coincide. The point A(5,4,-3) is on L1. Determine the coordinates of a point B on L2 such that vector AB is perpendicular to L2.

Homework Equations

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The Attempt at a Solution

I'm not sure where to start on this one. I know that eventually AB dot (2,1,-1) will equal 0, but I'm not really sure what to do with this question.

 

[tiny-tim]

Welcome to PF!

Hi adrimare! Welcome to PF!

… I know that eventually AB dot (2,1,-1) will equal 0 …

That's right!

So just write B in terms of t, and chug away.

 

[adrimare]

How do you do that? That's my main source of confusion here. How to write B in terms of t?

 

[Mark44]

Every point on L₂ has to satisfy L₂'s equation, which means that every point on L₂ has coordinates of (2 + 2t, 1 + t, 2 -t) for some value of t.

 

[adrimare]

So I do (5,4,-3)-(2+2t,2+t,2-t) to get AB and then do AB dot (2,1,-1) and find t somehow?

 

[Mark44]

Almost. AB = <2 + 2t, 1[/color] + t, 2 - t> - <5, 4, -3>. The dot product of AB and <2, 1, -1> should be 0.

 

[adrimare]

Great! Thanks! I got it! I have a sort of general question for you that I don't think is worthy of its own thread really. I was just wondering what makes a line parallel to a plane?

 

[Mark44]

A line is parallel to a plane if it (the line) is parallel to some line segment in the plane.

Another way to say this is that the line is parallel to the plane if it is perpendicular to the plane's normal.

 

[adrimare]

Thanks a lot! That helped with a big question! I'm starting a couple more threads with other questions if you want to look at them.