Vectors, Curves, and Surfaces in Space
Core definitions and formulas for R^n geometry, dot/cross products, lines, planes, and parametric curves.
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Questions Covered in This Set
10 cards to master
Give both forms of the dot product u·v.
Algebraic: Σ u_i v_i. Geometric: ‖u‖‖v‖cosθ. The equality is why arithmetic can answer geometric questions (angles, lengths, perpendicularity).
Formula for the projection of u onto v
proj_v u = ((u·v)/(v·v)) v — the 'shadow' of u along v; the seed of least squares, Fourier series, and orthogonal decomposition.
What does the cross product u×v give you, and in what dimension does it live?
Only in R^3: a vector perpendicular to both u and v, with length ‖u‖‖v‖sinθ = area of the parallelogram they span. Computed as the symbolic determinant with i, j, k in the top row.
Two key algebraic properties of the cross product
Anticommutative: u×v = −(v×u). Not associative: (u×v)×w ≠ u×(v×w) in general.
What does the scalar triple product u·(v×w) measure?
The signed volume of the parallelepiped spanned by u, v, w — the same idea as the Jacobian determinant: determinants measure how volume scales.
Vector equations of a line and a plane in space
Line: r(t) = r₀ + t d (point plus direction). Plane: n·(r − r₀) = 0, i.e. a(x−x₀)+b(y−y₀)+c(z−z₀)=0.
How do you find the plane through three given points?
Form two edge vectors from the points, cross them to get the normal n, then use n·(r − r₀) = 0.
Distance from a point Q to a plane through P₀ with normal n
d = |n·P₀Q| / ‖n‖ — a projection of the displacement vector onto the unit normal.
For r(t) = (x(t), y(t), z(t)), what are r′(t), ‖r′(t)‖, and r″(t)?
Differentiate componentwise: r′ is velocity (tangent to the curve), ‖r′‖ is speed, r″ is acceleration.
Velocity and speed of the helix r(t) = (cos t, sin t, t)
r′(t) = (−sin t, cos t, 1), so speed = √(sin²t + cos²t + 1) = √2, constant. Arc length over [0, 2π] is 2π√2.
