Cheat Sheet – Vector Algebra
⃗ | cos θ
Dot Product: a⃗. ⃗b = |a⃗||b
Scalar Multiplication: λa⃗
Properties
Geometry
Geometry
λ=1
λ>1
0<λ<1
λ = −1
−1 < λ < 0
λ < −1
⃗ ⇔ a⃗ ∥ b
⃗ (collinear)
a⃗ = λb
(a⃗, ⃗b ≠ ⃗0)
⃗
Addition: a⃗ + b
⃗ =b
⃗ . a⃗
⃗a. b
a⃗. a⃗ = |a⃗|2
⃗ ) = λ(a⃗. ⃗b)
(λa⃗). ⃗b = a⃗. (λb
⃗ + c) = a⃗. ⃗b + a⃗. c
a⃗. (b
⃗|
Projection of a⃗ on ⃗b = a⃗. ⃗b/|b
⃗ = 0 ⇔ a⃗ ⊥ b
⃗ (a⃗, b
⃗ ≠0
⃗)
a⃗. b
Geometry
Parallelogram
Law
Properties
Triangle Law
⃗ | sin θ n̂
Cross Product: a⃗ × ⃗b = |a⃗||b
⃗ = −b
⃗ × a⃗
a⃗ × b
a⃗ × a⃗ = ⃗0
⃗ ) = λ(a⃗ × ⃗b)
(λa⃗) × ⃗b = a⃗ × (λb
⃗ + c) = a⃗ × b
⃗ + a⃗ × c
a⃗ × (b
a⃗ × ⃗b = 0 ⇔ a⃗ ∥ ⃗b (a⃗, ⃗b ≠ ⃗0)
Any vector ⊥ to both a⃗ and ⃗b is λ(a⃗ × ⃗b)
⃗|
Area of a parallelogram = |a⃗ × b
Area of a triangle = 1/2|a⃗ × ⃗b|
⃗ are the
⃗a, b
coterminous
edges
Properties
Geometry
a⃗ + ⃗b = ⃗b + a⃗
a⃗ + ⃗0 = a⃗
a⃗ + (−a⃗) = ⃗0
⃗ ) = ka⃗ + kb
⃗
k(a⃗ + b
⃗ + c) = (a⃗ + ⃗b) + c
a⃗ + (b
Linear Combination: λ1 ⃗⃗⃗
a1 + λ2 ⃗⃗⃗⃗
a2 + ⋯ + λn ⃗⃗⃗⃗
an
Geometry
⃗ c a⃗] = [c a⃗
[a⃗ ⃗b c] = [b
⃗ c] = [a⃗
[λa⃗ ⃗b c] = [a⃗ λb
⃗b]
⃗b λc] = λ[a⃗ ⃗b c]
⃗ ] = −[b
⃗ a⃗ c]
[a⃗ ⃗b c] = −[c ⃗b a⃗] = −[a⃗ c b
[a⃗ ⃗b c] = 0 if any two of a⃗, ⃗b, c are equal/parallel
Volume of a parallelepiped = [a⃗ ⃗b c] a⃗ ⃗b c are the
coterminous
1
Volume of a tetrahedron = 6 [a⃗ ⃗b c]
edges
⃗ c] = 0 ⇔ a⃗, b
⃗ , c are coplanar (a⃗, b
⃗ ,c ≠ 0
⃗)
[a⃗ b
⃗ × c) = (a⃗. c)b
⃗ − (a⃗. b
⃗ )c
Vector Triple Product: a⃗ × (b
Any vector coplanar with a⃗ and ⃗b is
⃗
λ1 a⃗ + λ2 b
⃗ ≠0
⃗)
⃗ ; a⃗ ∦ b
(a⃗, b
Geometry
Properties
Polygon Law
⃗ c] = a⃗. (b
⃗ × c)
Scalar Triple Product: [a⃗ b
⃗ × c) is coplanar with ⃗b and c
a⃗ × (b
and perpendicular to a⃗
Vector Quadruple Product
Any vector in space is
⃗ + λ3 c
λ1 a⃗ + λ2 b
⃗ . ⃗d) − (a⃗. ⃗d)(b
⃗ . c)
(a⃗ × ⃗b). (c × ⃗d) = (a⃗. c)(b
(a⃗, ⃗b, c ≠ ⃗0; a⃗, ⃗b, c are non-coplanar)
⃗ ) × (c × d
⃗ ) = [a⃗ b
⃗ d
⃗ ]c − [a⃗ b
⃗ c]d
⃗ = [c d
⃗ a⃗]b
⃗ − [c d
⃗ b
⃗ ]a⃗
(a⃗ × b
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