Geometric Bites

Converting the Vector Equation of a Line into Cartesian Form A straight line in three-dimensional space can be expressed using vectors. One important vector form is (𝐑 − 𝐀) × 𝐁 = 0 This equation states that the displacement vector from a fixed point 𝐀 to a general point 𝐑 is parallel to the direction vector 𝐁. Two non-zero vectors have a zero cross product precisely when they are parallel. From this fact, the Cartesian (symmetric) equation of the line can be derived. 1. Substituting Coordinate Vectors The general point on the line is written as 𝐑 = (x, y, z) The fixed point is 𝐀 = (x₁, y₁, z₁) The direction vector is 𝐁 = (l, m, n) Substituting these into the vector equation yields: ((x, y, z) − (x₁, y₁, z₁)) × (l, m, n) = 0 which simplifies to: (x − x₁, y − y₁, z − z₁) × (l, m, n) = 0 2. Using the Condition for a Zero Cross Product If two non-zero vectors have a zero cross product, then one is a scalar multiple of the other. T...

The Shortest Distance Between Two Skew Lines in ℝ³ This post derives, from first principles, a vector formula for the shortest distance between two skew lines in ℝ³. The argument uses only the definitions and basic properties of the dot product and cross product; no higher results are assumed. 1. Vector Equations of the Lines Let the two lines be given in vector form by r = a + λb r = c + μd where: a , c are position vectors of fixed points on each line, b , d are non-zero direction vectors, λ, μ ∈ ℝ are parameters. The lines are skew if they are neither parallel nor intersecting. Our goal is to find a closed-form expression for the minimum distance between them. 2. Direction of the Common Perpendicular The segment that realises the shortest distance lies along a line perpendicular to both direction vectors b and d . A vector perpendicular to both is given by their cross product: b × d. Assuming b and d are not parallel, b × d ≠ 0. A unit ...

Area of a Triangle in R3 in Terms of Vectors This post presents a complete derivation of the area of triangle ABC in R3 using vectors alone. The result follows directly from the definition of the cross product and basic algebraic properties. No geometric shortcuts or previously established vector identities are assumed. Every step is obtained from first principles. 1. Vector Representation of the Triangle Let A, B and C be points in R3 with position vectors A, B and C. Two sides of triangle ABC are represented by: B - A C - A If theta is the angle between these vectors, the classical formula for the area of the triangle is: Area = (1/2)*|B - A|*|C - A|*sin(theta) 2. Magnitude of the Cross Product For any vectors U and V in R3, the magnitude of the cross product is: |U x V| = |U|*|V|*sin(theta) Applying this to U = B - A and V = C - A gives: | (B - A) x (C - A) | = |B - A|*|C - A|*sin(theta) Substituting into the area formula yields: Area(ABC) = (1/2)*| (B ...

Full Coordinate Derivation of (B - A) x (C - A) in R3 This derivation shows every algebraic step involved in expanding the cross product (B - A) x (C - A) using only coordinates. No vector identities are assumed in advance. All identities that appear at the end arise directly from the coordinate formula and elementary algebra. This method provides complete transparency and is the foundation for many geometric and analytic results involving the cross product. 1. Vectors and Cross Product Formula A = (a1, a2, a3) B = (b1, b2, b3) C = (c1, c2, c3) For vectors U = (u1, u2, u3) and V = (v1, v2, v3), the cross product is defined in coordinates by: U x V = ( u2*v3 - u3*v2, u3*v1 - u1*v3, u1*v2 - u2*v1 ) This is the only formula used. Every identity later in the derivation follows from substituting coordinates into this definition. 2. Basic Cross Products Needed for the Expansion A x A A x A = ( a2*a3 - a3*a2, a3*a1 - a1*a3, a1*a2 - a2*a1 ) = (0, 0, 0) A vector...

🧭 A Primer for Cross Product Calculations Right Hand Rule for Cross Product. Source: https://commons.wikimedia.org/wiki/File:Right-hand_rule_for_cross_product.png 1. Basis Vectors î = (1, 0, 0), ĵ = (0, 1, 0), k̂ = (0, 0, 1) These are unit and mutually perpendicular vectors. 2. Dot Product (for reference) 𝐀 · 𝐁 = |𝐀| |𝐁| cos θ î · ĵ = ĵ · k̂ = k̂ · î = 0 î² = ĵ² = k̂² = 1 3. Definition of the Cross Product 𝐀 × 𝐁 = |𝐀| |𝐁| sin θ n̂ θ is the angle from 𝐀 to 𝐁. n̂ is a unit vector perpendicular to both 𝐀 and 𝐁. The direction of n̂ follows the right-hand rule (anticlockwise = positive, clockwise = negative). 4. Fundamental Basis Cross Products î × ĵ = k̂ ĵ × k̂ = î k̂ × î = ĵ Reversing the order changes the sign: ĵ × î = −k̂ k̂ × ĵ = −î î × k̂ = −ĵ And any vector crossed with itself is zero: î × î = ĵ × ĵ = k̂ × k̂ = 0 5. Expansion in Component Form 𝐀 = a₁ î + a₂ ĵ + a₃ k̂ 𝐁 = b₁ î + ...