3D Vectors and Skew Lines

Also called

skew lines · shortest distance between two lines · distance between skew lines · 3D vectors · three dimensional geometry · vector equation of a line · common perpendicular · cross product · dot product · coordinate axes 3D · parallel lines distance · intersecting lines 3D

Topics

• a right-handed frame: x right, y away, z up, with i x j = k
• a vector from its components: v = 2i + j + 1.5k
• vector addition head to tail and by the parallelogram
• dot product as the projection of one vector on another
• cross product: perpendicular to both vectors, right-hand rule
• vector equation of a line r = a + λb: a is a position vector, b a direction vector
• skew lines are neither parallel nor coplanar; a flat view can make them seem to cross
• the shortest segment between skew lines is perpendicular to both (r·b1 = 0, r·b2 = 0)
• d = |(a2 − a1)·(b1 × b2)| / |b1 × b2| for nonparallel lines
• intersecting lines: d = 0 and the lines share a plane
• parallel lines: b1 × b2 = 0, so use d = |(a2 − a1) × b1| / |b1|

About this model

A GLB file of about 424 KB, shown in Lably's 3D viewer and, on a phone, placed in your room with AR. Credit: Authored for Lably AR (newly made schematic maths geometry). | Original procedural geometry and animation authored for Lably; no third-party mesh is included. The two Tactiles coordinate-axis files (CC BY-SA 4.0) were inspected but not used.

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