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Coincident, Parallel, Intersecting & Skew Lines
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Classifying vector lines in 3D as coincident, parallel, intersecting or skew using direction vectors and point checks, with the angle between lines found by \(\theta=\cos^{-1}\left(\frac{\mathbf{b}_1\cdot\mathbf{b}_2}{\left|\mathbf{b}_1\right|\left|\mathbf{b}_2\right|}\right)\).
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Parallel lines in 3D
AHL 3.15
Two vector lines are parallel if their direction vectors are scalar multiples of each other and the lines are not the same.
Consider two lines:
r1=a1+λb1,r2=a2+μb2.
These lines are parallel if b1=kb2 for some scalar k, but a1 does not lie on r2.
Coincident lines
AHL 3.15
Two vector lines are coincident if they represent exactly the same line, meaning every point on one line also lies on the other. For lines given by:
r1=a1+λb1,r2=a2+μb2,
they are coincident if:
Their direction vectors are parallel, so b1=kb2.
A point from one line (e.g., a2) lies on the other line, satisfying a2=a1+λb1 for some scalar λ.
Intersecting lines
AHL 3.15
Two vector lines intersect if they share exactly one common point. That means they are not parallel.
Skew lines
AHL 3.15
Two lines in three-dimensional space are skew if they are neither parallel nor intersecting.
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