Vectors and Geometry in Space
Cheat sheet.

The key rules, formulas and reminders from all 4 topics, gathered into reference cards.

Key formulas, conditions and traps · Read down each column.

Vectors, Dot Products and Cross Products

$u\cdot v=\|u\|\|v\|\cos\theta$ , $\|u\times v\|=\|u\|\|v\|\sin\theta$ , and volume is $|u\cdot(v\times w)|$ .

The dot product is a scalar; the cross product is an oriented vector. A zero vector has no direction.

Line: $r=r_0+tv$ . Plane: $n\cdot(r-p)=0$ . Point-plane distance: $|n\cdot q-d|/\|n\|$ .

A line direction and a plane normal play different roles. Nonparallel lines in space need not intersect.

v=r',\quad a=r'',\quad \text{speed}=\|r'\|,\quad L=\int_a^b\|r'\|dt.

Constant speed can coexist with nonzero acceleration. A tangent line requires a nonzero velocity direction.

T=r'/\|r'\|,\quad \kappa=\|r'\times r''\|/\|r'\|^3,\quad a_T=d\|r'\|/dt,\quad a_N=\kappa\|r'\|^2.

Curvature is geometric and has inverse-length units. These formulas require nonzero speed; a normal direction may fail where curvature is zero.

2 reference blocks

$u\cdot v=\|u\|\|v\|\cos\theta$ , $\|u\times v\|=\|u\|\|v\|\sin\theta$ , and volume is $|u\cdot(v\times w)|$ .

The dot product is a scalar; the cross product is an oriented vector. A zero vector has no direction.

2 reference blocks

Line: $r=r_0+tv$ . Plane: $n\cdot(r-p)=0$ . Point-plane distance: $|n\cdot q-d|/\|n\|$ .

A line direction and a plane normal play different roles. Nonparallel lines in space need not intersect.

2 reference blocks

v=r',\quad a=r'',\quad \text{speed}=\|r'\|,\quad L=\int_a^b\|r'\|dt.

Constant speed can coexist with nonzero acceleration. A tangent line requires a nonzero velocity direction.

2 reference blocks

T=r'/\|r'\|,\quad \kappa=\|r'\times r''\|/\|r'\|^3,\quad a_T=d\|r'\|/dt,\quad a_N=\kappa\|r'\|^2.

Curvature is geometric and has inverse-length units. These formulas require nonzero speed; a normal direction may fail where curvature is zero.