Notes, summaries, assignments, exams, and problems for Mathematics

Finding the Vertex Coordinates of a Parallelogram

The points A (-2, 3, 1), B (2, -1, 3), and C (0, 1, -2) are consecutive vertices of the parallelogram ABCD.

(a) Find the Vertex Coordinates of D

If ABCD are the vertices of a parallelogram, free vectors AB and DC are equal:

• AB = (4, -4, 2)
• DC = (-x, 1 - y, -2 - z)

Equating coordinates, we have x = -4, y = 5, and z = -4. The missing point is D (-4, 5, -4).

(b) Equation of the Line Through B and Parallel to Diagonal AC

The line passes through point B (2, -1, 3) and has a direction vector AC = (2, -2, -3). Its continuous equation is:

(x - 2) / 2 = (y + 1) / -2 = (z - 3) / -3

(c) Equation of the Plane Containing the Parallelogram

We can use point B (2, -1, 3) and the vectors BA = (-4, 4, -2) and BC = (-2,... Continue reading "Solving Problems with Parallelograms, Lines, and Planes in 3D Space" »

Soil Volume Calculation for Excavation

This section details the calculation of soil volume extracted between two distinct profiles, a common task in civil engineering and surveying projects.

Profile Dimensions and Separation

• First Profile Surface Area (St): 32 m²
• Second Profile Cross-Section: A trapezoid with a height of 3m, a lower base of 6m, and an upper base of 17m.
• Distance Separating Profiles (d): 54m

Calculating the Second Profile's Surface Area (Sd)

The area of the trapezoidal second profile is calculated as:

Sd = [(Lower Base + Upper Base) / 2] × Height

Sd = [(6 + 17) / 2] × 3 = 34.5 m²

Calculating Partial Volumes (Vt and Vd)

Using a specific volume computation method for irregular shapes:

Vt = 0.5 × (St)² / (St + Sd) × d

Vt = 0.5 × (32)

1. Thales' Theorem and Similar Triangles

Thales' Theorem is fundamental in geometry. It states that if a straight line is drawn parallel to one side of a triangle, it divides the other two sides proportionally, creating a smaller triangle that is similar to the original one.

The theorem is valid only for similar triangles. For example, given a triangle ABC, if we trace a segment MN parallel to one side, we obtain a smaller triangle AMN similar to ABC.

Criteria for Triangle Similarity

Two triangles are considered similar if they satisfy any of the following premises:

• Their corresponding sides are proportional.
• They have two corresponding angles equal (which implies all three angles are equal).
• They have one equal angle, and the sides forming that angle

1. Payment Process Methods

Payment methods can be categorized as follows:

• 1) Cash: Includes physical cash income deposited into the supplier's account, bank transfers, Cheques, debit cards, and credit cards.
• 2) Deferred: Includes financial instruments like the Bill of Exchange and the Promissory Note.

2. The Receipt Document

A receipt is a document issued by the person who collects the money and delivered to the payer as proof of payment for acquired goods or services provided. Receipts must be printed, typically using numbered receipt books (with a matrix or stub). The person paying must be given their receipt, while the matrix (stub) is held by the issuer to justify the payment.

3. The Cheque (Check)

A cheque is a document issued against a bank... Continue reading "Financial Instruments: Cheques, Bills, Notes, and Cards" »

Working Capital

Working Capital measures the capacity for payment in the ordinary course of business activity. It's calculated as: Current Assets (CA) - Current Liabilities (CL)

• CA > CL: Positive Working Capital. The business has the potential for investment. Working Capital should never exceed 10% of CA, as these are idle funds.
• CA < CL: Negative Working Capital. This may indicate a suspension of payments or insolvency. It usually signifies mismanagement in the negotiation of ordinary business activity, but it doesn't always mean a bad situation.

Acid Test

The Acid Test measures a company's capacity to meet all of its short-term debts. It's calculated as: (Current Liabilities - Treasury) / Available. This indicates immediate liquidity;... Continue reading "Understanding Key Financial Ratios for Businesses" »

Equivalence of Decimals

Decimals are the expression of decimal fractions. We can obtain equivalent fractions by multiplying the numerator and denominator by 10, 100, etc. To compare two decimal numbers, it is sufficient to compare their corresponding decimal fractions.

Equivalence of Percentages

For example, 2/5 is equivalent to 40/100 (or 40%). Frequent problems involve scenarios where an object's price is increased or decreased, asking for:

• The original price
• The final price after the rebate
• The percentage of the discount

The Measurement Problem in Math Education

The traditional learning of mathematics has often been rigid. This proposes a new method that stimulates thought through trial and error. Learning to measure magnitudes is often identified... Continue reading "Effective Math Teaching Strategies for Deeper Understanding" »

Quadratic Forms and Symmetric Matrices

The quadratic form is defined as w(x) = x^TAx, where M(w)_c = 1/2(A + A^T) is the associated symmetric matrix in the canonical basis.

Change of basis formula: M(w)_b = (P_BC)^T · M(w)_c · P_BC.

Rank properties: rg(w) = rg(A) = rg(M(w)_c). If rg(w) = max, the form is non-degenerate; otherwise, it is degenerate.

Diagonalization of Symmetric Matrices

For a symmetric matrix A, applying elementary row and column operations leads to (D | P^T), where D is diagonal. The goal is to obtain zeros in the off-diagonal positions.

A form w admits a non-degenerate orthonormal basis if the rank is maximum or the diagonal elements of D are non-zero.

Signature and Sylvester's Criterion

The signature sig(w) = sig(A) = (p, r_p), where rg(w)

Fundamentals of Statistical Analysis

Population and Sampling

All statistical study refers to a set of elements called the population. When the population is very large, and the goal is to focus on only a part of it, a sample is taken. This sample can represent the entire population or be a partial survey.

Analyzing Representative Samples

A representative sample can be analyzed using two main methods based on strata (subgroups) of the population:

• Equal Allocation: The sample is taken such that the number of elements selected from each stratum of the population is the same.
• Proportional Allocation: The representative sample is taken in proportion to the number of elements in each stratum.

Characteristics and Variables

The characteristics (or qualities)... Continue reading "Statistical Concepts: Sampling, Variables, and Data Visualization" »

Financial Mathematics

Core Definitions

• Annuity: A series of equal payments made at regular intervals to pay off a debt or build a fund.
• Capital: The principal amount of money agreed upon for an investment or loan.
• Interest: The amount earned or paid for the use of money, directly proportional to capital, time, and the interest rate.
• Compound Interest: Interest calculated on the initial principal and also on the accumulated interest of previous periods; its growth is exponential.
• Simple Interest: Interest calculated only on the initial principal; the capital remains constant each period, resulting in arithmetic growth.
• Amount: The total sum of money to be paid or received at the end of an agreed period, calculated as capital plus interest.
• Deadline: