Cartesian Coordinates Quiz

Study Notes

Cartesian Coordinates

Definition: A coordinate system that specifies each point uniquely in a plane by a pair of numerical coordinates, which are the signed distances to two fixed perpendicular directed lines (axes).
Components:
- Axes: Two perpendicular lines:
  - x-axis: Horizontal line.
  - y-axis: Vertical line.
- Origin: The point where the axes intersect, denoted as (0,0).
Coordinates:
- Each point is represented as an ordered pair (x, y):
  - x: Horizontal distance from the y-axis.
  - y: Vertical distance from the x-axis.
Quadrants:
- The Cartesian plane is divided into four quadrants:
  1. Quadrant I: (x > 0, y > 0)
  2. Quadrant II: (x < 0, y > 0)
  3. Quadrant III: (x < 0, y < 0)
  4. Quadrant IV: (x > 0, y < 0)
Distance Formula:
- The distance ( d ) between two points ( (x_1, y_1) ) and ( (x_2, y_2) ):
  - ( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} )
Midpoint Formula:
- The midpoint ( M ) between two points ( (x_1, y_1) ) and ( (x_2, y_2) ):
  - ( M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) )
Slope of a Line:
- The slope ( m ) between two points ( (x_1, y_1) ) and ( (x_2, y_2) ):
  - ( m = \frac{y_2 - y_1}{x_2 - x_1} )
Equation of a Line:
- In slope-intercept form:
  - ( y = mx + b )
    - where ( m ) is the slope and ( b ) is the y-intercept.
- In point-slope form:
  - ( y - y_1 = m(x - x_1) )
Applications:
- Used in physics to describe motion, forces, and other phenomena in a two-dimensional space.
- Basis for more complex systems like polar coordinates and three-dimensional Cartesian coordinates.

Cartesian Coordinates

Coordinate system that uniquely defines points in a plane through ordered pairs of numerical coordinates.
Axes consist of two fixed perpendicular lines:
- x-axis: Represents horizontal distances.
- y-axis: Represents vertical distances.
Origin: Intersection point of axes, represented as (0,0).

Coordinates

Each point expressed as (x, y):
- x indicates distance from the y-axis.
- y indicates distance from the x-axis.

Quadrants

The Cartesian plane is divided into four quadrants:
- Quadrant I: Both x and y are positive (x > 0, y > 0).
- Quadrant II: x is negative, y is positive (x < 0, y > 0).
- Quadrant III: Both x and y are negative (x < 0, y < 0).
- Quadrant IV: x is positive, y is negative (x > 0, y < 0).

Distance Formula

Calculates the distance ( d ) between points ( (x_1, y_1) ) and ( (x_2, y_2) ):
- Formula: ( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} ).

Midpoint Formula

Determines the midpoint ( M ) between two points ( (x_1, y_1) ) and ( (x_2, y_2) ):
- Formula: ( M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) ).

Slope of a Line

Measures the slope ( m ) between two points:
- Formula: ( m = \frac{y_2 - y_1}{x_2 - x_1} ).

Equation of a Line

Slope-intercept form: Represents the equation as ( y = mx + b ), where ( m ) is the slope and ( b ) is the y-intercept.
Point-slope form: Written as ( y - y_1 = m(x - x_1) ).

Applications

Cartesian coordinates are essential in physics to model two-dimensional motion and force dynamics.
Foundation for more advanced systems like polar coordinates and three-dimensional Cartesian coordinates.

Description

Test your understanding of Cartesian coordinates, including their definitions, components, and quadrants. This quiz covers the essential concepts of the coordinate system and the distance formula used to calculate the distance between points in a plane.