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Vectors - Coggle Diagram
Vectors
Trigonometry
Branch of mathematics that studies the relationships between sides and angles of triangles.
Trigonometric functions
Sine (sin) = opposite leg / hypotenuse
Cosine (cos) = adjacent leg / hypotenuse
Tangent (tan) = opposite leg / adjacent leg
Reciprocal functions
Cosecant (csc)
Secant (sec)
Cotangent (cot)
Angles
Measured in degrees or radians.
Conversion: 360° = 2π radians.
Trigonometric graphs
They represent periodic functions (sine, cosine, tangent).
Features: amplitude, period, maximum/minimum points and intersection.
Geometric Vectors
Definition
Quantity that has magnitude and direction
Representation
Arrow → sense = direction, length = magnitude
Magnitude
|v| = length of the vector.
Direction
Given by an angle with respect to an axis or coordinates
Vectors in three-dimensional space
They are used to represent displacements, forces or velocities.
Represented by components:
v = (x, y, z)
Basic Operations
Addition: placing vectors end to end → resulting vector.
Subtraction: place in opposite direction → vector from the end of the second to the beginning of the first.
Scalar multiplication: changes the length, not the direction.
Dot product (v·w): sum of products of components → scalar result.
Applications
Describing lines and planes in analytic geometry:
Line: point + direction vector.
Plane: point + normal vector.
Components in R2 and R3
R² (plane): v = (x, y)
R³ (space): v = (x, y, z)
Operations
Addition: (x₁ + x₂, y₁ + y₂)
Subtraction: (x₁ − x₂, y₁ − y₂)
Scalar product: k·v = (k·x, k·y [, k·z])
Examples:
u = (2, 3), v = (-1, 5) → u + v = (1, 8)
u = (1, 2, 3), v = (-2, 4, 1) → u + v = (-1, 6, 4)
Analytical Applications:
Line equation: point + direction vector.
Plane equation: point + normal vector.
Scalar product
Operation between two vectors → scalar result.
Formulas:
v · w = ||v|| ||w|| cos(θ)
v · w = v₁w₁ + v₂w₂ + … + vₙwₙ
Example:
v = (2, 3), w = (4, 1) → v·w = 2×4 + 3×1 = 11
Geometric interpretation
If v·w > 0 → acute angle
If v·w = 0 → perpendicular vectors
If v·w < 0 → obtuse angle
Applications
Vector projection
Calculation of physical work (F·d)
Determine orthogonality
Norm of a vector
Measure of the length or magnitude of a vector.
General formula:
||v|| = √(v₁² + v₂² + … + vₙ²)
Example:
v = (3, 4) → ||v|| = √(3² + 4²) = 5
Properties:
||v|| ≥ 0
||v|| = 0 ⇔ v = (0, 0, …, 0)
Uses:
Calculating distances between points.
Physics: magnitude of force, velocity, or displacement.
Operations: Analytical and Graphical Form
Analytical form: Uses components (x, y, z).
Graphical form: Uses arrows in the plane or space.
Addition
Analytics: (x₁ + x₂, y₁ + y₂)
Graphics: Place vectors end-to-end.
Subtraction
Analytics: (x₁ − x₂, y₁ − y₂)
Graph: Reverse the direction of the second vector.
Scalar multiplication
Analytics: k·v = (k·x, k·y)
Graph: lengthen or shorten the arrow without changing direction.
Dot product
Analytics: v·w = v₁w₁ + v₂w₂ + …
Graph: v·w = ||v|| ||w|| cos(θ)
Graphic representation
Vectors are drawn as arrows:
Start point → end point.
Length = magnitude.
Direction = orientation
In R²: Cartesian plane.
In R³: three-dimensional space.
Addition and subtraction graphically
Addition → triangle or parallelogram method.
Subtraction → reverse direction of the second vector.
Advantages: Facilitates the geometric visualization of magnitude, direction and operations.