Geometric Bites

Parabolas in sight: The Clifton Suspension Bridge, Bristol, United Kingdom. A quadratic function is a function whose graph is a parabola (a U-shaped curve). One of the most useful ways to write a quadratic is in vertex form , because it shows the parabola’s turning point immediately. 1) The vertex form A quadratic function in vertex form is written as: f(x) = a(x - h) 2 + k This form is especially helpful because the values h and k tell you the vertex directly. 2) The vertex (turning point) The vertex is the point where the parabola changes direction. In vertex form: Vertex = (h, k) If the parabola opens up , the vertex is the lowest point (a minimum). If the parabola opens down , the vertex is the highest point (a maximum). 3) What the number a does The number a controls two key things: the direction the parabola opens, and how wide or narrow it is. a > 0 means the parabola opens up (U-shape). a < 0 means the parabola opens dow...

1) n! (factorial) Meaning: tells you how many permutations (complete orderings) you can generate with n objects at your disposal. Definition: n! = n × (n−1) × (n−2) × … × 2 × 1, with 0! = 1. Example: 5! = 5×4×3×2×1 = 120. 2) n!/(n−r)! (permutations of r choices from n; order matters) Meaning: tells you how many ordered outcomes you can generate when you make r choices out of a collection of n objects, without reuse. How to see it: 1st choice: n options 2nd choice: (n−1) options 3rd choice: (n−2) options … rth choice: (n−r+1) options Multiply: n × (n−1) × … × (n−r+1) = n!/(n−r)!. Example (n=5, r=2): 5P2 = 5!/(5−2)! = 5!/3! = (5×4×3×2×1)/(3×2×1) = 5×4 = 20. 3) (n!/(n−r)!)/r! = n!/((n−r)! r!) (combinations; order neglected) Meaning: tells you how many selections you can make when choosing r objects from n, where order does not matter. ...

🧭 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 + ...