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 September 8th, 2010, 07:05 AM #1 Member   Joined: Apr 2010 Posts: 91 Thanks: 0 Vectors-intersection of lines Find the intersection of the lines: $(7+2\lambda,3,-17-7\lambda): \lambda \epsilon \mathbb{R}$ and $(3 -4\lambda ,3,-3+14\lambda) : \lambda\epsilon \mathbb{R}$ I tried equating the x, y and z coordinates of both lines, but when I try to solve for lambda in each I dont get the same answer, meaning they dont intersect. However the answer to this problem was the line: $(1+2\lambda , 3,4-7\lambda):\lambda \epsilon \mathbb{R}$. I dont understand how "a line" could be the answer.
 September 8th, 2010, 07:34 AM #2 Global Moderator     Joined: Nov 2009 From: Northwest Arkansas Posts: 2,766 Thanks: 4 Re: Vectors-intersection of lines There's definitely something wrong here!
September 8th, 2010, 03:51 PM   #3
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Re: Vectors-intersection of lines

Quote:
 Originally Posted by TsAmE Find the intersection of the lines: $(7+2\lambda,3,-17-7\lambda): \lambda \epsilon \mathbb{R}$ and $(3 -4\lambda ,3,-3+14\lambda) : \lambda\epsilon \mathbb{R}$ I tried equating the x, y and z coordinates of both lines, but when I try to solve for lambda in each I dont get the same answer, meaning they dont intersect. However the answer to this problem was the line: $(1+2\lambda , 3,4-7\lambda):\lambda \epsilon \mathbb{R}$. I dont understand how "a line" could be the answer.
Typical in problems like this, each line would have a different parameter.
7+2u=3-4v (x equality)
-17-7u=-3+14v (z equality)
Solve for u and v to get point of intersection.

September 8th, 2010, 08:29 PM   #4
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Re: Vectors-intersection of lines

Hello, TsAmE!

[color=green]mathman[/color] is correct . . . Two parameters should have been used.

Quote:
 $\text{Find the intersection of the lines: }\;\begin{Bmatrix}L_1: && (7\,+\,2\lambda,\:3,\:-17\,-\,7\lambda) &\;& \lambda\,\in\,\mathbb{R} \\ \\ \\ L_2: && (3\, \,-4\mu,\:3,\:-3\,+\,14\mu) &\;& \mu\,\in\,\mathbb{R} \end{Bmatrix}$

$\text{The first line }L_1\text{ contains }P(7,\:3,\:-17)\text{ and has direction vector }\vec u \:=\:\langle 2,\:0,\:-7\rangle$

$\text{The second line }L_2\text{ contains }Q(3,\:3,\:-3)\text{ and has direction vector }\vec v \:=\:\langle -4,\:0,\:14\rangle$

$\text{We see that: }\:\vec v \:=\:\langle -4,\:0,\:14\rangle \;=\;-2\,\cdot\,\langle 2,\:0,\:-7\rangle \;=\;-2\,\cdot\,\vec u$

[color=beige]. . [/color]$\text{Hence, the two lines are parallel.}$

$\text{We find that point }Q\text{ lies on line }L_1.$

[color=beige]. . [/color]$\text{Therefore, the two lines coincide . . . They "intersect everywhere".}$

September 9th, 2010, 02:34 AM   #5
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Re: Vectors-intersection of lines

I am a bit confused. Even though the 2 lines are parallel, howcome Q lies on the other line? Isnt it possible that if L1 is parallel to L2, Q might not lie on L1 (like this in attachment)?
Attached Images
 Lines.png (7.4 KB, 130 views)

 September 9th, 2010, 05:15 AM #6 Global Moderator     Joined: Nov 2009 From: Northwest Arkansas Posts: 2,766 Thanks: 4 Re: Vectors-intersection of lines It is possible that parallel lines are not the same. But once we determined that they have (at least) ONE point in common, then they must have ALL points in common.
 September 9th, 2010, 02:03 PM #7 Member   Joined: Apr 2010 Posts: 91 Thanks: 0 Re: Vectors-intersection of lines Oh ok. How would the answer: $(1+2\lambda , 3,4-7\lambda):\lambda \epsilon \mathbb{R}$ represent that these lines lie on each other?
 September 9th, 2010, 02:22 PM #8 Global Moderator     Joined: Nov 2009 From: Northwest Arkansas Posts: 2,766 Thanks: 4 Re: Vectors-intersection of lines The format of the answer tells us about the question. If two or more lines cross (touch) in a single point, they intersect and the point is the solution to the system of equations. If two lines DON'T cross, then they are parallel (or skew, in higher dimensions). If two lines cross EVERYWHERE, then they are the same line. You can conclude that lines with the same slope through a common point are actually the same line. All that to say... if the ANSWER is a line and the QUESTION is about two lines, then the "two" lines are actually just the same line.

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