Problem 4 - Technical Mathematics with Calculus

Question

Directed Distance Find the directed distance \(A B\). $$A(-9,-2) ; B(17,-2)$$

Step-by-Step Solution

Answer

The directed distance AB is 26 units.

Step 1: Understand Directed Distance

Directed distance is a measure of the distance between two points along a directed line segment from one point to the other. In this case, the direction is from point A to point B, which implies that if B is to the right of A on the number line, the directed distance will be positive, and if it is to the left, it will be negative.

Step 2: Identify Coordinates

Identify the x-coordinates of points A and B. Point A has an x-coordinate of -9 and point B has an x-coordinate of 17.

Step 3: Calculate Directed Distance

Since the y-coordinates of A and B are the same, the directed distance AB can be found by subtracting the x-coordinate of A from the x-coordinate of B: \[ AB = x_B - x_A = 17 - (-9) = 17 + 9 = 26 \]Thus, the directed distance AB is 26 units.

Key Concepts

Coordinate Geometry

Coordinate geometry is a branch of mathematics that allows us to accurately describe the location of points, lines, and shapes within a given space. To make this possible, we use a coordinate system, such as the Cartesian coordinate system, which is the most common. This system uses two axes, the horizontal x-axis and the vertical y-axis, to define a point in a plane with a pair of numbers known as coordinates.

In our exercise, the points A with coordinates (-9, -2) and B (17, -2) lay on this plane. Essentially, coordinate geometry transforms algebraic equations into visual representations, enabling us to analyze and solve geometric problems using algebraic methods.

Distance Formula

The distance formula in coordinate geometry is a vital tool for finding the length of a line segment connecting two points. It stems from the Pythagorean theorem for right-angled triangles and allows us to calculate the distance 'D' between two points (A and B) with coordinates (Ax, Ay) and (Bx, By) using the formula: \[ D = \sqrt{(Bx - Ax)^2 + (By - Ay)^2} \]In simpler terms, we're finding the hypotenuse of a right triangle where the other two sides are the differences between the x and y coordinates of the points. In the context of our given problem, since the y-coordinates of points A and B are the same, our distance formula reduces to the absolute difference between the x-coordinates of A and B only.

Directed Line Segment

A directed line segment differs from a simple line segment in that it has a specific orientation from one endpoint to the other. It is essentially a vector that has both magnitude (length) and direction. When we discuss directed distances, we are concerned with both the length and the direction (from point A to point B or vice versa).

In the problem provided, the directed distance from point A to point B is simply the difference in their x-coordinates since the y-coordinates are the same. The direction here is important: if we were asked for the directed distance from point B to point A, the answer would be negative, since point A is to the left of point B on the x-axis. Understanding the concept of directed line segments is crucial for interpreting problems in vector calculus and physics, where direction and magnitude play pivotal roles.