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A Distance

Lingual Usage

  1. (countable) The amount of space between two points, usually geographical points, usually (but not necessarily) measured along a straight line.
  2. Length or interval of time.
  3. (countable, informal) The difference; the subjective measure between two quantities.
  4. Remoteness of place; a remote place.
  5. Remoteness in succession or relation.
  6. A space marked out in the last part of a racecourse.
  7. (uncountable, figuratively) The entire amount of progress to an objective.
  8. (uncountable, figuratively) A withholding of intimacy; alienation; variance.
  9. The remoteness or reserve which respect requires; hence, respect; ceremoniousness.
  10. The space measured back from the winning-post which a racehorse running in a heat must reach when the winner has covered the whole course, in order to run in the final heat.
Synonyms

Mathematics

Analytic Geometry

Distance between two Points:

Two points \(P_1(x_1,y_1,z_1)\), \(P_2(x_2,y_2,z_2)\):

\[\bar{P_1P_2}=\sqrt {(x_2-x_1)^2+(y_2-y_1)^2+(z_2-z_1)^2}\]

Distance between a point and a line:

The distance d between a point \(P_0(x_0,y_0)\) and a line \(\Delta; ax + by + c = 0\) in the Cartesian plane can be found using the formula:

\[d = \frac{\left|ax_0 + by_0 + c\right|}{\sqrt{a^2 + b^2}}\]

Here, a, b, and c are the coefficients of the equation of the line, and ((x_0, y_0)) are the coordinates of the point.

Alternatively, the equation of the line in the form (y = mx + b) (where (m) is the slope and (b) is the y-intercept). The distance ((d)) between a point ((x_0, y_0)) and the line (y = mx + b) is given by:

\[d = \frac{\left|mx_0 - y_0 + b\right|}{\sqrt{m^2 + 1}}\]

These formulas are derived from the geometric interpretation of the distance between a point and a line. The numerator of each fraction represents the signed perpendicular distance between the point and the line, and the denominator normalizes this distance. The absolute value is taken to ensure a positive distance regardless of the side of the line on which the point lies.