What is a Cartesian Vector?

  • A Cartesian vector is a way to represent geometric quantities with both magnitude (length) and direction in a coordinate system, usually 2D or 3D space.
  • It uses an ordered set of numbers, called components, to indicate the vector's displacement along each axis of the coordinate system.

Example in 2D

  • Consider a vector pointing from the origin (0, 0) to the point (3, 4) on a 2D Cartesian plane:
    • Components: This vector can be represented as (3, 4).
    • X-Component: 3 indicates the displacement along the X-axis.
    • Y-Component: 4 indicates the displacement along the Y-axis.

Visual Representation

Cartesian vectors are commonly visualized as arrows:

  • Tail: The starting point of the arrow is usually placed at the origin.
  • Head: The arrowhead points in the direction of the vector.
  • Length: The arrow's length represents the vector's magnitude.

3D Cartesian Vectors

  • In 3D, a Cartesian vector has three components (x, y, z), representing displacement along the X, Y, and Z axes.
  • Example: Vector from (0, 0, 0) to (2, -5, 3) is represented as (2, -5, 3).

Common Operations with Cartesian Vectors

  • Addition & Subtraction: Done component-wise.
  • Scalar Multiplication: Multiply each component by a scalar value to change the magnitude.
  • Magnitude (Length): Calculated using the Pythagorean theorem (or distance formula) in multi-dimensions.
  • Dot Product: Measures the angle between two vectors.
  • Cross Product (3D): Produces a vector perpendicular to two original vectors.

Applications

Cartesian vectors are fundamental in:

  • Physics: Representing forces, velocities, accelerations, etc.
  • Engineering: Analyzing structures, fluid flow, and other physical systems.
  • Computer Graphics: Manipulating objects and positions in 2D and 3D environments.
  • Mathematics: Abstract vector spaces and linear algebra.

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