3D Geometry¶
Recall (doc/postdoc). Not a first course.
Job: affine geometry of \(\mathbb{R}^3\); lines/planes via linear algebra; cross product areas.
Core¶
| Object | Description |
|---|---|
| Line | \(r(t)=r_0+td\), \(d\neq0\) |
| Plane | \(n\cdot(r-r_0)=0\) or \(ax+by+cz=d\) |
| Distance pt–plane | $\dfrac{ |
| Skew lines | nonparallel, nonintersecting (not coplanar) |
Cross product: \(u\times v\perp u,v\); \(\|u\times v\|=\) parallelogram area; scalar triple \([u,v,w]=u\cdot(v\times w)=\) parallelepiped volume.
Dihedral angle: angle between normals of two planes.
Used later for¶
- Vectors: \(\times\), projections
- Multivariable calculus: surfaces, flux
- Differential geometry: frames in \(\mathbb{R}^3\)
- PDE / physics math: div/grad/curl identities
Check¶
Can two lines in \(\mathbb{R}^3\) fail to intersect yet not be parallel?
Yes — skew lines. In \(\mathbb{R}^2\), nonparallel lines always meet (Euclidean/affine plane).