Vector Basics and Operations

This page introduces fundamental concepts and operations related to vectors in mathematics and physics.

Współrzędne wektora (Vector coordinates) are typically represented as ordered pairs (x, y) in 2D space or triples (x, y, z) in 3D space. These coordinates define the vector's position relative to the origin.

Definition: A vector is a quantity that has both magnitude and direction, represented by an arrow in geometric representations.

Długość wektora (Vector length) is calculated using the Pythagorean theorem. For a vector AB with coordinates [x, y], the length is given by:

|AB| = √(x² + y²)

Example: For a vector v = [3, 4], its length would be √(3² + 4²) = √25 = 5.

The concept of opposite vectors is introduced. For a vector v = [a, b], its opposite vector is -v = [-a, -b].

Highlight: Opposite vectors have the same magnitude but point in opposite directions.

Dodawanie wektorów (Vector addition) is performed by adding corresponding components. For vectors v = [a, b] and w = [c, d], their sum is:

v + w = [a+c, b+d]

Example: If v = [2, 3] and w = [1, 2], then v + w = [2+1, 3+2] = [3, 5].

Vector subtraction follows a similar principle, subtracting corresponding components:

v - w = [a-c, b-d]

Example: If v = [3, 5] and w = [-3, -2], then v - w = [3-(-3), 5-(-2)] = [6, 7].

The page concludes with an introduction to the Iloczyn skalarny wektorów (Dot product of vectors). This operation results in a scalar value and is calculated by multiplying corresponding components and summing the results.

Vocabulary: The dot product, also known as the scalar product, is an algebraic operation that takes two vectors and returns a single scalar value.