Problem 6 - Technical Mathematics with Calculus

Question

Directed Distance Find the directed distance \(A B\). $$B(11.5,3.68) ; A(11.5,-5.38)$$

Step-by-Step Solution

Answer

The directed distance from point A to point B is 9.06.

Step 1: Understanding Directed Distance

Directed distance between two points is the distance with a sign that indicates direction. Since points A and B have the same x-coordinate, the directed distance will be vertical (along the y-axis).

Step 2: Calculate the Difference in y-coordinates

To find the directed distance between points A and B along the y-axis, we subtract the y-coordinate of A from the y-coordinate of B. So, the calculation is B's y-coordinate minus A's y-coordinate: \(3.68 - (-5.38)\).

Step 3: Perform the Subtraction

Subtracting the y-coordinate of A from B gives us the directed distance: \(3.68 - (-5.38) = 3.68 + 5.38\).

Step 4: Calculate the Sum

Adding the two values we get the directed distance: \(3.68 + 5.38 = 9.06\). The result is positive, which means the direction is from A to B upwards on the y-axis.

Key Concepts

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is a branch of mathematics that allows us to analyze geometric shapes through a coordinate system. This system uses ordered pairs of numbers, known as coordinates, to represent points on a plane. Each point is determined by its position along the horizontal x-axis and the vertical y-axis.

In the context of the exercise where we find the directed distance between points A and B, understanding the coordinates of each point is crucial. A coordinate pair like \(11.5,-5.38\) signifies the position of a point, where 11.5 is the distance along the x-axis, and -5.38 is the distance along the y-axis. Here, a negative y-coordinate implies that the point lies below the origin along the y-axis.

Distance Calculation

The calculation of distance in coordinate geometry can often be approached using the Pythagorean theorem when dealing with horizontal or vertical distances, or the distance formula for any other pair of points. For direct vertical or horizontal distances, the process is simplified because one coordinate value remains constant while the other changes.

In this case, with points A and B having the same x-coordinate \(11.5\), the only changing value is the y-coordinate. By focusing solely on the y-coordinates, we avoid the complexity of the full distance formula and provide a simpler calculation. Just subtracting one y-coordinate from the other gives the distance between the two points along the y-axis.

Vertical Distance

Vertical distance refers to the measure of distance between two points that lie directly above or below each other on the coordinate plane. It's a concept that plays a key role when two points share the same x-coordinate, as seen in the directed distance problem. In these situations, the directed distance is simply the absolute difference between the y-coordinates.

However, in directed distance calculations, unlike the absolute distance, we do not ignore the signs. The positive or negative result tells us in which direction the distance is 'directed'. In the given solution, since the directed distance is positive (\(9.06\)), it indicates that if we start at point A, we must move upwards along the y-axis to reach point B.