Problem 3 - Technical Mathematics with Calculus

Question

If \(h\) and \(k\) are positive quantities, in which quadrants would the following points lie? $$(-h, k)$$

Step-by-Step Solution

Answer

The point \

Step 1: Understanding the Coordinates System

The coordinate system is divided into four quadrants. The signs of the coordinates determine the quadrant. Quadrant I (positive x, positive y), Quadrant II (negative x, positive y), Quadrant III (negative x, negative y), Quadrant IV (positive x, negative y).

Step 2: Identifying the Quadrant for (-h, k)

Given the point (-h, k), where h and k are positive, the x-coordinate is negative and the y-coordinate is positive. According to the signs of the coordinates, this point lies in Quadrant II.

Key Concepts

Understanding the Coordinate System

The coordinate system, also known as the Cartesian plane, is a fundamental concept in mathematics and physics. It's a two-dimensional plane where each point is defined by a pair of numbers, usually referred to as coordinates. Think of it as a map that can show the location of points in space using two reference lines that intersect at a right angle.

The point where these two lines meet is called the origin, and from there, we can measure distances along the horizontal and vertical axes, which are traditionally labeled as the x-axis and y-axis, respectively. Any point on this plane can be represented by an ordered pair \(x, y\).

The coordinate system is divided into four regions, known as quadrants, and they are numbered counterclockwise starting from the upper right corner. To identify the quadrant of a point, we check the signs of its x and y coordinates:

Exploring the Cartesian Plane

The Cartesian plane, named after the French mathematician René Descartes, is the grid-like framework on which the coordinate system is laid out. It’s like a canvas where we plot our numerical data. The plane is divided into four quadrants by the intersection of the x-axis and y-axis.

Each quadrant contains points with specific sign combinations:

• Quadrant I: Both x and y coordinates are positive.
• Quadrant II: x is negative, y is positive.
• Quadrant III: Both x and y coordinates are negative.
• Quadrant IV: x is positive, y is negative.

Points on the axes themselves are not considered to be in any quadrant. A key to mastering the Cartesian plane is to understand these sign patterns, as they allow you to quickly determine the position of any point just by looking at its coordinates.

Interpreting Positive and Negative Coordinates

In the coordinate system, positive and negative coordinates reveal not just the magnitude of a point's distance from the origin but also its direction relative to the x-axis and y-axis. Here's what you need to know:

Positive x-coordinates indicate a point is to the right of the y-axis, while negative x-coordinates show it's to the left. Similarly, positive y-coordinates mean the point is above the x-axis, and negative y-coordinates mean it's below. Therefore, when we look at a point like \( -h, k \) where h and k are both positive quantities, we interpret \( -h \) as a movement to the left of the origin on the x-axis and \( k \) as a movement upwards from the origin on the y-axis. Combining these directional movements, we can conclude that the point \( -h, k \) is in Quadrant II, where all points have negative x and positive y coordinates.