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The distance formula is a mathematical tool used to determine the space between two points in a coordinate plane. This formula is particularly useful in geometry and is often applied to find the length of a line segment between two points, say points P and Q, with coordinates (x1, y1) and (x2, y2).
The formula is essentially:

• \( \|PQ\| = \sqrt{(x2-x1)^2 + (y2-y1)^2} \)

This might look complicated but don't worry; it's simpler than it seems.
The expression inside the square root \((x2-x1)^2 + (y2-y1)^2\) essentially calculates the horizontal and vertical distances between these points, then combines them.
The result is a single number that represents the distance, ensuring you consider both the horizontal and vertical gaps simultaneously.

The distance formula is actually derived from the Pythagorean theorem, which is a fundamental principle in trigonometry.
The Pythagorean theorem states that in a right-angled triangle, the square of the hypotenuse (longest side) is equal to the sum of the squares of the other two sides.
Mathematically, this is expressed as:

• \( c^2 = a^2 + b^2 \)

The distance between two points in a plane can be visualized as the hypotenuse of a right-angled triangle, where the horizontal and vertical lines from P to Q form the other two sides.
By using the coordinates of P and Q, you effectively create this triangle, making you utilize the theorem to find the distance.

Hence, when you use \( \|PQ\| = \sqrt{(x2-x1)^2 + (y2-y1)^2} \), you're applying the Pythagorean theorem with coordinates as legs of the triangle.

A directed line segment is a concept that involves both the magnitude (or length) and the direction from one point to another.
Unlike an ordinary line segment which only considers extent, a directed line segment also incorporates the direction from start to endpoints.
This means it is represented with an arrow, which gives it a direction, sometimes referred to as a vector.
For example, the directed line segment \( \overrightarrow{PQ} \) considers P as the starting point and Q as the endpoint, effectively creating a sense of direction from P to Q.
To calculate its magnitude, that is, the straight-line distance from P to Q, you would use the distance formula. Although direction is a vector aspect, magnitude uses a scalar measurement provided by the formula.
In various applications, understanding the direction is crucial, particularly in physics and engineering, but in pure distance calculations, the focus is primarily on length.