Geometric Bites

The area of a parallelogram can be found using two vectors. Let the two vectors be: A and B If A is the base of the parallelogram, then the height is the perpendicular part of B. From the diagram: height = |B|sin(θ) Therefore: Area = base × height Area = |A||B|sin(θ) This is the standard formula for the area of a parallelogram formed by two vectors. Removing the Angle The formula Area = |A||B|sin(θ) uses the angle θ between the two vectors. But the angle is not always given. To remove the angle, start with the dot product identity: A · B = |A||B|cos(θ) Now square both sides: (A · B) 2 = |A| 2 |B| 2 cos 2 (θ) Using the trigonometric identity: cos 2 (θ) = 1 − sin 2 (θ) we get: (A · B) 2 = |A| 2 |B| 2 (1 − sin 2 (θ)) Expand the right-hand side: (A · B) 2 = |A| 2 |B| 2 − |A| 2 |B| 2 sin 2 (θ) Now rearrange: |A| 2 |B| 2 sin 2 (θ) = |A| 2 |B| 2 − (A · B) 2 The Area Identity Since Area = |A||B|sin(θ) s...

Two of the most important results in differential calculus are d/dx(ln x) = 1/x d/dx(aˣ) = aˣ ln a. These formulas are closely connected. One describes the derivative of the natural logarithm, while the other gives the derivative of an exponential function with positive base. Together, they reveal the deep relationship between logarithms, exponentials, inverse functions, and differentiation. 1. Derivative of the natural logarithm Begin with ln x = y. This is equivalent to eʸ = x. Differentiate both sides with respect to y: dx/dy = eʸ. Now invert this result: dy/dx = 1/eʸ. Since eʸ = x, substitute back: dy/dx = 1/x. Therefore, d/dx(ln x) = 1/x, for x > 0. So the derivative of the natural logarithm is the reciprocal of x. 2. Derivative of the exponential function aˣ Now let y = aˣ, where a > 0 and a ≠ 1. Take logarithms to base a: logₐ y = x. Now apply the change-of-...

Proving that the sum of three consecutive integers is divisible by 3. Algebraic proof questions in Edexcel International GCSE (Higher) often look difficult because they are written in words. The quickest way to handle them is to translate the words into standard algebraic forms that guarantee the number property you need (even, odd, multiple, consecutive, square, etc.). Once the translation is correct, the rest of the proof is usually straightforward simplification, factoring, and a clear final statement. This post gives a compact “toolkit” of the most common forms, presented in a table you can reuse, plus a small set of extras and techniques that frequently appear in Higher-tier proof questions. The core principle In an algebraic proof, represent the numbers so the required property is built in. For example: If a number is even, write it as 2n for some integer n. If a number is a multiple of 3, write it as 3n for some integer n. The phrase “for some integer n” matters...

An improper fraction being turned into a mixed number. Fractions are used to represent parts of a whole, but once a fraction becomes larger than 1, there are two common ways to write it: as an improper fraction or as a mixed number . These two forms often represent exactly the same quantity; the difference is simply how the number is written. Improper fractions An improper fraction is a single fraction where the numerator is greater than or equal to the denominator. This means the fraction is at least 1 whole (and possibly more). Examples: 7/4 12/5 9/9 In 7/4, there are 7 parts, and each whole is made from 4 parts. Since 7 is larger than 4, the value is greater than 1. Mixed numbers A mixed number is written as a whole number followed by a proper fraction. The proper fraction shows the leftover part after counting whole units. Examples: 1 3/4 2 2/5 3 1/6 In 1 3/4, the “1” shows one whole, and “3/4” shows three extra quarters. Same value, differe...

Structure: The Great Pyramid of Giza, Egypt. Source:  https://commons.wikimedia.org/wiki/File:Kheops-Pyramid.jpg When physicists and mathematicians talk about the “architecture” of the universe, it can sound like they mean the universe was built like a house. Most of the time, they do not mean that. In this context, “architecture” is a practical word for something simpler and more specific: The universe behaves as if it has deep, consistent structure — stable rules and constraints that keep working wherever we look. This post explains why many scientists think that is a reasonable conclusion, without assuming any advanced mathematics. 1) The universe repeats itself: patterns that do not go away In everyday life, you already rely on the universe being stable: If you drop a mug, it falls. If you heat water, it boils at roughly the same temperature (given the same pressure). If you shine light through glass, it bends in predictable ways. Science extends this ...

A quadratic function and its turning point. Link to graph:  https://www.desmos.com/calculator/fktyfs12st A quadratic function is any function of the form f(x) = ax² + bx + c with a ≠ 0 . Its graph is a parabola, and every parabola has exactly one turning point (also called the vertex ). Completing the square is fundamental because it rewrites the quadratic as a shifted square , which makes the turning point immediately visible. Vertex form: the turning point is built in The vertex form of a quadratic is: f(x) = a(x − h)² + k This form is powerful because it exposes two facts at once: (x − h)² ≥ 0 for all real x (a square is never negative). (x − h)² = 0 happens exactly when x = h . So: If a > 0 , then a(x − h)² ≥ 0 , so the smallest possible value of f(x) is k , achieved at x = h (a minimum). If a < 0 , then a(x − h)² ≤ 0 , so the largest possible value of f(x) is k , achieved at x = h (a maximum). Therefore, in vertex form, the turning ...