Introduction to Coordinate Calculation

In coordinate geometry, determining the coordinates of a point that lies at a specific fraction of the distance between two given points is a common task. This article will guide you through the process of finding the coordinates of a point that is ( frac{2}{3} ) of the distance from point ( P ) to point ( Q ). We will explore the mathematical formula and provide examples in one, two, and three dimensions.

Mathematical Formulation

To find the coordinates of a point that is ( frac{2}{3} ) of the distance from point ( P ) to point ( Q ), you can use the formula for finding a point that divides a line segment in a given ratio. The formula for a point ( R ) that divides the segment ( PQ ) in the ratio ( m : n ) is:

Rleft( frac{mx_2 nx_1}{m n}, frac{my_2 ny_1}{m n} right)

For our case, where ( m 2 ) and ( n 1 ) (corresponding to ( frac{2}{3} ) of the distance), the formula simplifies to:

Rleft( frac{2x_2 x_1}{3}, frac{2y_2 y_1}{3} right)

Example Calculation

Let's consider points ( P(1, 2) ) and ( Q(4, 6) ).

Steps:

Substitute the coordinates of points ( P ) and ( Q ) into the simplified formula: Rleft( frac{2 cdot 4 1 cdot 1}{3}, frac{2 cdot 6 1 cdot 2}{3} right) Simplify the values inside the parentheses: Rleft( frac{8 1}{3}, frac{12 2}{3} right) Rleft( frac{9}{3}, frac{14}{3} right) The result is: Rleft( 3, frac{14}{3} right)

Therefore, point ( R ) that is ( frac{2}{3} ) of the distance from ( P ) to ( Q ) is ( left( 3, frac{14}{3} right) ).

When working in one dimension, points are represented as a single coordinate, such as ( P(x_1) ) and ( Q(x_2) ). The formula simplifies to:

R P frac{2}{3} (Q - P) frac{1}{3}P frac{2}{3}Q

In two or three dimensions, the process is the same, but each component is calculated separately.

For points ( P (x_1, y_1, z_1, ldots) ) and ( Q (x_2, y_2, z_2, ldots) ), the coordinates of the point ( R ) that is ( frac{2}{3} ) of the distance from ( P ) to ( Q ) are given by:

R (x_1 frac{2}{3}(x_2 - x_1), y_1 frac{2}{3}(y_2 - y_1), z_1 frac{2}{3}(z_2 - z_1), ldots)

To find the specific point ( R ) between any two points ( P ) and ( Q ), simply substitute their coordinates into the appropriate formula. This method is highly versatile and can be applied in one, two, or more dimensions, making it a valuable tool in coordinate geometry.