Geometric Bites

A skew-symmetric matrix is a square matrix whose transpose equals its negative. The construction below begins with an arbitrary 3 × 3 matrix, subtracts its transpose, and derives the complete general form without assuming the result in advance. K is skew-symmetric precisely when Kᵀ = −K. 1. Begin with an arbitrary matrix Let A = a b c d e f g h i and Aᵀ = a d g b e h c f i Transposition reflects the entries across the main diagonal: rows become columns and columns become rows. 2. Subtract the transpose Define K = A − Aᵀ. Subtract corresponding entries: K = a−a b−d c−g d−b e−e f−h g−c h−f i−i The diagonal entries cancel: ...

A matrix transforms input vectors into output vectors. Some independent directions survive, some are combined, and some may disappear completely into the zero vector. Span, dimension, column space, rank, null space and nullity describe this process precisely. input space → matrix transformation → column space 1. Span: all reachable combinations Given vectors v₁, v₂, …, vₖ, their span is the set of every vector that can be made by scaling and adding them: Span{v₁, v₂, …, vₖ} = {c₁v₁ + c₂v₂ + ⋯ + cₖvₖ : c₁, c₂, …, cₖ ∈ ℝ}. One non-zero direction spans a line through the origin. Two independent directions span a plane through the origin. Three independent directions in ℝ³ span all of ℝ³. 2. Independence and dimension Vectors are linearly independent when none of them can be constructed from the others. Each independent vector contributes a genuinely new direction. A dependent vector contributes no new...

Let v = (p, q, r) T be a vector in ℝ³. From its three components, form the 3 × 3 skew-symmetric matrix K = ( 0 −r q r 0 −p −q p 0 ) . The entries reflected across the main diagonal have opposite signs, while every diagonal entry is zero. Therefore, K T = −K. Multiplication by the vector Matrix-vector multiplication can be written as a linear combination of the columns of K. The first column is multiplied by p, the second by q, and the third by r: K v = p ( 0 r −q ) + q ( −r 0 p ) + r ( q −p 0 ) . Distributing p, q and r gives K v = ( 0 · p + (−r) · q + q · r r · p + 0 · q + (−p) · r −q · p + p · q + 0 · r ) . Combining the entries in each row produces ...

This proof demonstrates that subtracting the transpose of any real 3 × 3 matrix from the original matrix produces a 3 × 3 skew-symmetric matrix. Construction Let M = ( a b c d e f g h i ) Its transpose is M T = ( a d g b e h c f i ) Subtracting the transpose from the original matrix gives K = M − M T = ( 0 b − d c − g d − b 0 f − h g − c h − f 0 ) ...

Direction Cosines of a Unit Vector A vector in 3D can be written as v = (x, y, z). This vector points from the origin to the point (x, y, z). Its direction depends on how much it travels in the x-direction, the y-direction and the z-direction. Magnitude of the Vector The magnitude, or length, of v is |v| = √(x² + y² + z²). This comes from the 3D version of Pythagoras' theorem. The vector has three perpendicular components: x, y and z. Squaring them, adding them, and taking the square root gives the total length. Unit Vector A unit vector is a vector with length 1. To turn v into a unit vector, divide every component by the magnitude of v: v̂ = (1 / |v|)(x, y, z). So v̂ = (x / |v|, y / |v|, z / |v|). This new vector points in the same direction as v, but its length is exactly 1. The Dot Product The dot product has two important forms. Algebraic form: a · b = a₁b₁ + a₂b₂ + a₃b₃. Geometric form: a · b = |a||b|cos(θ). The algebraic form uses co...

The associative, commutative and distributive laws are three of the most important structural rules in algebra. They explain how expressions may be grouped, reordered, expanded and simplified without changing their mathematical value. These laws are used throughout arithmetic, algebra, factorisation, equation solving and mathematical proof. Associative Law The associative law describes how terms may be grouped when the same operation is repeated. If an operation is associative, changing the placement of the brackets does not change the final value of the expression. For addition: a + (b + c) = (a + b) + c For example: 1 + (2 + 3) = (1 + 2) + 3 The associative law also applies to multiplication: a × (b × c) = (a × b) × c For example: 2 × (3 × 4) = (2 × 3) × 4 Subtraction is not associative because changing the grouping can change the result. a − (b − c) ≠ (a − b) − c For example: 1 − (2 − 3) ≠ (1 − 2) − 3 Commutative Law The commutative law describe...

These workings use compound angle identities to derive double angle, triple angle, and related trigonometric identities. Compound Angles, Extras 1. Deriving sin(2θ) sin(θ + θ) = sinθ cosθ + cosθ sinθ = 2sinθ cosθ = sin(2θ) 2. Deriving cos(2θ) cos(θ + θ) = cosθ cosθ - sinθ sinθ = cos 2 θ - sin 2 θ = cos(2θ) 3. Deriving cos(2θ) = 2cos²θ - 1 cos(2θ) = cos 2 θ - sin 2 θ = cos 2 θ - (1 - cos 2 θ) = cos 2 θ - 1 + cos 2 θ = 2cos 2 θ - 1 4. Deriving cos(2θ) = 1 - 2sin²θ cos(2θ) = cos 2 θ - sin 2 θ = 1 - sin 2 θ - sin 2 θ = 1 - 2sin 2 θ 5. Deriving sin(3θ) sin(2θ + θ) = sin(2θ)cosθ + cos(2θ)sinθ = 2sinθ cosθ cosθ + (1 - 2sin 2 θ)sinθ = 2sinθ cos 2 θ + sinθ - 2sin 3 θ = sinθ(2cos 2 θ + 1) - 2sin 3 θ = sinθ(2(1 - sin 2 θ) + 1) - 2sin 3 θ = sinθ(2 - 2sin 2 θ + 1) - 2sin 3 θ = sinθ(3 - 2sin 2 θ) - 2sin 3 θ = 3sinθ - 2sin 3 θ - 2sin 3 θ = 3sinθ - 4sin 3 θ 6. Deriving cos(3θ) cos(2θ + θ) = cos2θ cosθ - sin2...