Vectors and Geometry

Active Calculus · CC BY-SA · activecalculus.org

Scalars measure magnitude. Vectors measure magnitude and direction. The dot product measures alignment, the cross product produces a perpendicular vector, and together they give you the geometry of lines and planes in R3.

Vectors in R2 and R3

A vector is an ordered list of components. Addition is component-wise. Scalar multiplication scales each component. The magnitude (length) is the square root of the sum of squared components.

Scheme

; Vectors as lists
(define (vec-add u v) (map + u v))
(define (vec-scale c v) (map (lambda (x) (* c x)) v))
(define (vec-mag v) (sqrt (apply + (map (lambda (x) (* x x)) v))))

(define u (list 3 4))
(define v (list 1 -2))

(display "u = ") (display u) (newline)
(display "v = ") (display v) (newline)
(display "u + v = ") (display (vec-add u v)) (newline)
(display "2*u = ") (display (vec-scale 2 u)) (newline)
(display "|u| = ") (display (vec-mag u)) (newline)

; 3D vectors
(define a (list 1 2 3))
(define b (list 4 5 6))
(display "a + b = ") (display (vec-add a b)) (newline)
(display "|a| = ") (display (vec-mag a))

Dot product

The dot product u · v = sum of component products = |u||v|cos(theta). It measures how much two vectors point in the same direction. Zero means perpendicular.

Scheme

; Dot product
(define (dot u v) (apply + (map * u v)))
(define (vec-mag v) (sqrt (apply + (map (lambda (x) (* x x)) v))))

(define u (list 1 2 3))
(define v (list 4 -5 6))

(display "u . v = ") (display (dot u v)) (newline)
; 1*4 + 2*(-5) + 3*6 = 4 - 10 + 18 = 12

; Angle between vectors: cos(theta) = (u.v) / (|u|*|v|)
(define cos-theta (/ (dot u v) (* (vec-mag u) (vec-mag v))))
(display "cos(theta) = ") (display cos-theta) (newline)
(display "theta (radians) = ") (display (acos cos-theta)) (newline)

; Perpendicular check
(define p (list 1 0))
(define q (list 0 1))
(display "p . q = ") (display (dot p q)) (display " (perpendicular)")

Cross product

The cross product u x v is defined only in R3. It produces a vector perpendicular to both u and v, with magnitude |u||v|sin(theta). Direction follows the right-hand rule.

Scheme

; Cross product in R3
(define (cross u v)
  (list
    (- (* (list-ref u 1) (list-ref v 2))
       (* (list-ref u 2) (list-ref v 1)))
    (- (* (list-ref u 2) (list-ref v 0))
       (* (list-ref u 0) (list-ref v 2)))
    (- (* (list-ref u 0) (list-ref v 1))
       (* (list-ref u 1) (list-ref v 0)))))

(define (dot u v) (apply + (map * u v)))
(define (vec-mag v) (sqrt (apply + (map (lambda (x) (* x x)) v))))

(define u (list 1 2 3))
(define v (list 4 5 6))
(define w (cross u v))

(display "u x v = ") (display w) (newline)
; (2*6-3*5, 3*4-1*6, 1*5-2*4) = (-3, 6, -3)

; Verify perpendicularity
(display "w . u = ") (display (dot w u)) (newline)   ; 0
(display "w . v = ") (display (dot w v)) (newline)   ; 0
(display "|u x v| = ") (display (vec-mag w))

Lines and planes

A line through point P in direction d: r(t) = P + t*d. A plane through point P with normal n: n · (r − P) = 0. The cross product of two direction vectors in the plane gives the normal.

Scheme

; Line: r(t) = P + t*d
(define P (list 1 2 3))
(define d (list 2 -1 4))

(define (point-on-line t)
  (map (lambda (p di) (+ p (* t di))) P d))

(display "Line: r(t) = (1,2,3) + t*(2,-1,4)") (newline)
(display "t=0: ") (display (point-on-line 0)) (newline)
(display "t=1: ") (display (point-on-line 1)) (newline)
(display "t=2: ") (display (point-on-line 2)) (newline)

; Plane: n . (r - P) = 0
; Normal n = (1, 1, 1), point P = (0, 0, 0)
; Equation: x + y + z = 0
(define n (list 1 1 1))
(define (on-plane? point)
  (= 0 (apply + (map * n point))))

(display "Is (1,-1,0) on x+y+z=0? ") (display (on-plane? (list 1 -1 0))) (newline)
(display "Is (1,1,1) on x+y+z=0? ") (display (on-plane? (list 1 1 1)))

Notation reference

Symbol	Meaning
u · v	Dot product (scalar)
u × v	Cross product (vector, R3 only)
\|v\|	Magnitude / length
r(t) = P + td	Parametric line
n · (r − P) = 0	Equation of a plane

Neighbors

Calculus sequence

Linear Algebra Ch.2 — vector spaces formalize these operations
📐 Linear Algebra Ch.1 — vectors and dot products are the algebraic counterpart to geometric vectors
📐 Geometry Ch.2 — vectors in Euclidean space generalize to curved geometry
⚛ Physics Ch.1 — vectors as displacements, velocities, and forces

Foundations (Wikipedia)

← Applications of Integrals by june.kim Partial Derivatives →