What is skewness measure between two perpendicular lines?

Answer 1

Two lines that are perpendicular to the same [third] line can meet at the same point, be parallel to one another or be skew. If you are not sure about that, see below for examples of all three cases.

The skewness between the two perpendicular lines is the angle between the projection of one of the lines on the other.

In vector analysis, if the direction vectors of the two perpendicular lines are a and b, then if x is the angle between them,

cos(x) = a.b/(|ab|)

where a.b is the scalar or dot product of aand b and,

|a| and |b| are the magnitudes (lengths) of the two vectors.

x is a measure of the skewness.

Example:

Imagine yourself in a cuboid room facing one of the walls. The line where the floor meets the opposite wall is the reference line.

First consider the line where the left wall meets the floor and where the left wall meets the wall you're facing. Both these are perpendicular to the reference line. They meet: at the bottom-left-front corner of the room.

Second, consider the line where the left wall meets the floor and where the right wall meets the floor. Both these are also perpendicular to the reference line. They never meet: they are parallel.

Third, consider the line where the left wall meets the floor and the diagonal on the facing wall: from the top-left-front to the bottom-right-front. Again both these are perpendicular to the reference line. They are not parallel but they never meet either: they are skew.