Cartesian Coordinates Quiz

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.