Question

Directed Distance Find the directed distance \(A B\). $$B(-6,-6) ; A(-6,-7)$$

Short answer

The directed distance AB is 1 unit.

Step-by-step solution

Understand Directed Distance

Directed distance between two points in a coordinate plane is the distance with the specification of the direction from one point to another. If you are moving vertically, the directed distance can be found by subtracting the y-coordinates, while if moving horizontally, subtract the x-coordinates of the points.

Identify the Direction of Movement

Since the x-coordinates of the points A and B are the same (-6), the movement is vertical, along the y-axis. We will thus subtract the y-coordinate of A from that of B to find the directed distance AB.

Calculate the Directed Distance

Subtract the y-coordinate of A (-7) from that of B (-6): Directed distance AB = y-coordinate of B - y-coordinate of A = (-6) - (-7) = (-6) + 7 = 1. Thus, the directed distance AB is 1 unit. Note that because the result is positive, the direction is from A to B and upwards along the y-axis given the negative to positive orientation.

Key concepts

Coordinate Plane

The coordinate plane is a two-dimensional surface where each point is defined by a pair of numbers known as coordinates. These numbers correspond to the horizontal and vertical positions, respectively, on the plane. A coordinate plane is divided into four quadrants by the x-axis (horizontal line) and y-axis (vertical line), which intersect at the origin, marked as (0,0).

Points on the coordinate plane are written as \( (x, y) \) where \( x \) represents their horizontal distance from the origin, and \( y \) represents their vertical distance. Understanding the coordinate plane is crucial as it is the foundation for graphing equations, determining the location of points, and calculating distances between points.

Subtracting Coordinates

Subtracting coordinates enables us to find the distance and direction from one point to another on the coordinate plane. When subtracting coordinates, it's important to carefully consider the signs (+ or -) of the values you're working with.

To subtract coordinates and find the directed distance between two points, simply subtract the corresponding x or y values, depending on whether you're measuring horizontal or vertical distance. For example, if you have two points \( A(x_1, y_1) \) and \( B(x_2, y_2) \) and you want to find the vertical distance from \( A \) to \( B \) you would calculate \( B_y - A_y \) where \( B_y \) and \( A_y \) are the y-coordinates of points \( B \) and \( A \) respectively.

Vertical Movement

Vertical movement in the context of a coordinate plane refers to motion along the y-axis. This vertical motion is either upwards or downwards relative to the origin's position. When the y-coordinate increases, the movement is upwards, and when it decreases, it's downwards.

Assessing vertical movement is essential, especially when determining the directed distance between points that have the same x-coordinates. This scenario indicates that there has been no horizontal movement, and the directed distance is purely vertical. Always remember that the result of subtracting y-coordinates will indicate the direction: a positive result points upwards, and a negative result points downwards on the axis.