How to Calculate Distance & Displacement Between Coordinates
The distance between two points in space may be loosely described as the length of a straight line between those two points. The location of a point is given by its coordinates, which may be used to calculate the distance between the points. These coordinates are a set of numbers such that each number identifies the displacement of the point from an origin with respect to a particular dimension.
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Instructions
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1
Determine the distance between two points in a one dimensional space. Let point p have a coordinate p1 so that p = (p1) and let point q have a coordinate q1 so that q = (q1). The distance between the two points is given as d(p,q) = |p1 -- q1|. The absolute value of the difference between the coordinates is used because distance is always positive.
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2
Calculate the distance between two points in two dimensions. The point p will have the coordinates p1 and p2 such that p = (p1,p2). The point q will have the coordinates q1 and q2 such that q = (q1,q2).
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3
Let the points p and q be the end points of the hypotenuse of a right triangle. Using the results from steps 1 and 2, the lengths of the other two sides in the right triangle may be given as |p1 -- q1| and |p2 -- q2|.
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4
Use the Pythagorean theorem to calculate the distance between the points p and q. This theorem states that the length of a right triangle's hypotenuse is equal to the square root of the sum of the squares of the other two sides of the triangles. Thus, c = (a^2 + b^2)^(1/2) where c is the hypotenuse of the triangle, with a and b being the other two sides.
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Determine the length of the hypotenuse of the right triangle with the end points p and q. Using the Pythagorean formula c = (a^2 + b^2)^(1/2) and making the substitutions a = |p1 -- q1| and b = |p2 -- q2| gives the result c = ((p1 -- q1)^2 + (p2 -- q2)^2)^(1/2). Since the length of the hypotenuse c is the distance d(p,q), then d(p,q) = ((p1 -- q1)^2 + (p2 -- q2)^2)^(1/2).
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