Geometry Fundamentals: Triangles and Vector Analysis

Understanding Triangles and Their Properties

A triangle is a polygon with three sides. It is determined by three line segments called sides or three non-aligned points called vertices.

Key Characteristics of Triangles

• Plane figures: They exist in a two-dimensional plane.
• Area and Volume: They have area but no volume.
• Polygons: Triangles are classified as polygons.
• Interior Angles: The sum of its interior angles always equals 180°.

Classification of Triangles

Classification by Sides

• Equilateral: All three sides measure the same length.
• Isosceles: Two sides measure the same length.
• Scalene: All sides have different lengths.

Classification by Angles

• Right-angled: Contains one right angle (90°).
• Obtuse: One angle is obtuse (greater than 90°) and the other two angles are acute (less than 90°).
• Acute-angled: When all three angles are less than 90°.

Methods to Solve Triangles

Pythagoras' Theorem: In any right triangle, the square of the length of the hypotenuse equals the sum of the squares of the lengths of the legs.

The Law of Cosines: The cosine theorem allows us to solve triangles when the following are known:

• Two sides and the angle between them.
• Three sides.

Introduction to Vectors and Scalars

A vector is a straight and directed segment that has an origin and an end.

Scalars and Vector Magnitudes

Scalars: These are quantities that are completely specified by a number followed by a unit. Examples of scalars include length, temperature, mass, density, time, volume, and surface area.

Vector Magnitudes: These feature a module (magnitude) and require a specific direction. Vector quantities include force, velocity, displacement, momentum, and acceleration.

Different Classes of Vectors

• Fixed or Related Vectors: Those that have a fixed point of application in space.
• Sliding Vectors: Vectors whose point of application can be moved over the line of action where they are supported.
• Free Vectors: The set of vectors that have the same direction, magnitude, and sense but different lines of action.
• Polar Vectors: Those whose magnitudes represent a translation.
• Axial Vectors: Those whose magnitudes represent a rotation.
• Opposing Vectors: Two vectors having the same module and direction but opposite sense.
• Unit Vectors: Dimensionless vectors whose modulus is equal to the unit, used to specify a given direction.
• Parallel Vectors: Two vectors that have the same direction as their magnitudes are proportional.
• Basic Aspects (Basis Vectors): Unit vectors (magnitude equal to 1) whose directions and senses align with the coordinate axes.

Direction of a Vector in the Plane

The direction is the angle formed with respect to a reference axis. One method used for the direction of a vector refers to conventional directions: North, South, East, and West.