Common perpendicular of two lines

In summary, the common perpendicular of two lines is a line that is perpendicular to both lines and intersects them at their point of intersection. To find the common perpendicular of two lines, the slope-intercept form and point-slope form of the lines can be used to find the equation of the common perpendicular line. Two lines can only have one common perpendicular, and it may not always be in the same direction as the lines. The common perpendicular is useful in geometry for finding the shortest distance between two lines, and it is also used in other fields such as engineering and architecture. It is a fundamental concept in vector algebra and linear algebra.
  • #1
ydan87
25
0
Hi there,
The two given lines:
1) x / 2 = (y-1) / 2 = z
2) x + 1 = y – 2 = (z + 4) / 2
I need to find the common perpendicular between them.
Though when I tried to calculate the distance between them, I get 0.

Can someone please help?

Thanks in advance
 
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  • #2
Think about the cross product of their direction vectors.
 
  • #3
Easier than I thought! Thanks :)
 
  • #4
Direction vectors of lines 1 and 2: V1 = (2, 2, 1), V2 = (1, 1, 2)
V1 X V2 = (3, -3, 0)

Is it correct?
 
  • #5
Yes, it is.
 

1. What is the common perpendicular of two lines?

The common perpendicular of two lines is a line that is perpendicular to both lines and intersects them at their point of intersection.

2. How can I find the common perpendicular of two lines?

To find the common perpendicular of two lines, you can use the slope-intercept form of the lines and solve for the point of intersection. Then, using the point-slope form, you can find the equation of the common perpendicular line.

3. Can two lines have more than one common perpendicular?

No, two lines can only have one common perpendicular. If two lines are perpendicular to each other, they will have infinitely many common perpendiculars.

4. Is the common perpendicular always in the same direction as the lines?

No, the common perpendicular may not always be in the same direction as the lines. For example, if the lines are parallel, the common perpendicular will be perpendicular to both lines, but in a different direction.

5. How is the common perpendicular useful in geometry and other fields?

The common perpendicular is useful in geometry for finding the shortest distance between two lines. It is also used in fields such as engineering and architecture for constructing perpendicular structures and determining optimal placement of objects. Additionally, it is a fundamental concept in vector algebra and linear algebra.

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