Finding Equation of an Orthogonal Line

In summary, the conversation discusses finding the line passing through and orthogonal to two given lines, as well as the distance between the two lines. One of the participants is unsure about how to find a point on the line and another participant suggests using the distance calculation.
  • #1
TranscendArcu
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0

Homework Statement


Let L1 be the line (0,4,5) + <1,2,-1>t and L2 be the line (-10,9,17) + <-11,3,1>t

a) Find the line L passing through and orthogonal to L1 and L2
b) What is the distance between L1 and L2

The Attempt at a Solution


I only know how to do part of part a). I can only find the direction vector of the orthogonal line by taking the cross product. I have,

<1,2,-1> x <-11,3,1> = <5,10,25>, which I simplify to <1,2,5>.

It doesn't seem very obvious to me how I can find a point (presumably on either L1 or L2) such that a line containing this point, pointing in the direction of <1,2,5>, passes through both L1 and L2.

For part b), I presume that these two lines are skew. (How do you check if lines are parallel or intersect?) The distance between these two lines is

|<5,10,25> dot [(0,4,5) - (-10,9,17)]| = |<5,10,25> dot <10,-5,-12>| = |50 - 50 - 300| = 300. So the distance is 300?
 
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  • #2
Actually, I think I did the distance calculation wrong. What I actually have is 60/sqrt(30)
 
  • #3
TranscendArcu said:
Actually, I think I did the distance calculation wrong. What I actually have is 60/sqrt(30)

Looks good.

[itex]\displaystyle \frac{60}{\sqrt{30}}=2\sqrt{30}\,[/itex]
 
  • #4
Okay, well it's good to know I can do part b. But what I think I really don't understand is how to do part a. One of my friends said it could be done if one has already calculated the distance, but this seems like doing the problem backwards, which I would like to avoid if possible.
 

FAQ: Finding Equation of an Orthogonal Line

1. What is an orthogonal line?

An orthogonal line is a line that intersects another line at a right angle, or 90 degrees. This means that the two lines are perpendicular to each other.

2. How do you find the equation of an orthogonal line?

To find the equation of an orthogonal line, you first need to find the slope of the original line. Then, take the negative reciprocal of that slope and plug it into the slope-intercept form of a line, y = mx + b. This will give you the equation of the orthogonal line.

3. Can there be more than one orthogonal line for a given line?

Yes, there can be an infinite number of orthogonal lines for a given line. This is because any line that intersects the original line at a right angle can be considered an orthogonal line.

4. How does the equation of an orthogonal line relate to the original line?

The equation of an orthogonal line is perpendicular to the original line. This means that their slopes are negative reciprocals of each other, and they intersect at a right angle.

5. Are there any real-life applications of finding equations of orthogonal lines?

Yes, finding equations of orthogonal lines is commonly used in fields such as engineering and architecture. It is also used in mathematics to solve problems involving perpendicular lines and angles.

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