Are Skew Lines Considered Perpendicular in 3D with Parallel Vectors?

[mohamed el teir]

in 3D assume two skew lines L1: x=x°+at, y=y°+bt, z=z°+ct, L2: x=x°°+ds, y=y°°+es, z=z°°+fs. therefore L1, L2 parallel vectors are respectively: v1= <a, b, c>, v2= <d, e, f>. if v1.v2= ad+be+cf= 0 (vectors are perpendicular), are the line L1, L2 considered perpendicular also or beside the dot product of their parallel vectors being zero the lines must also intersect (must not be skew) ??

 

[RUber]

The lines would be considered orthogonal, but that does not imply that they intersect.
Consider the x and y axes. The intersect in the plane z=0. But if you raise the x-axis to y=0, z=1, it will never intersect the y-axis defined by x=0, z=0.

 

[aikismos]

https://en.wikipedia.org/wiki/Skew_lines

That article has some tests.

 

[mohamed el teir]

The lines would be considered orthogonal, but that does not imply that they intersect.
Consider the x and y axes. The intersect in the plane z=0. But if you raise the x-axis to y=0, z=1, it will never intersect the y-axis defined by x=0, z=0.

do you mean that in the case that x-axis is in its original position the x and y axes are perpendicular and intersect, and in the case that the x-axis is in y = 0, z = 1 the x and y axes are perpendicular and doesn't intersect ?

 

[aikismos]

do you mean that in the case that x-axis is in its original position the x and y axes are perpendicular and intersect, and in the case that the x-axis is in y = 0, z = 1 the x and y axes are perpendicular and doesn't intersect ?

Yes. Perpendicular lines only intersect if they lie in the same plane.

 

[RUber]

Yes. That was your question, right? If orthogonal lines must intersect or not.

 

[mohamed el teir]

Yes. That was your question, right? If orthogonal lines must intersect or not.

yes sort of that, if lines with perpendicular directions but the lines themselves are skew, are they considered perpendicular or not, so they are perpendicular, thank you