Problem 37 - Thomas Calculus

Question

A particle starts at \(A(-2,3)\) and its coordinates change by increments \(\Delta x=5, \Delta y=-6 .\) Find its new position.

Step-by-Step Solution

Answer

The new position is \((3, -3)\).

Step 1: Identifying Initial Position

The initial position given for the particle is the point \(A(-2, 3)\). This means the particle starts at \(x = -2\) and \(y = 3\).

Step 2: Understanding the Increments

The increments tell us how much to add or subtract from the initial coordinates: \(\Delta x = 5\) and \(\Delta y = -6\). This means we will move 5 units to the right in the \(x\)-direction and 6 units down in the \(y\)-direction.

Step 3: Calculating New x-coordinate

To find the new \(x\)-coordinate, add the increment \(\Delta x\) to the initial \(x\)-coordinate: \(x_{new} = -2 + 5 = 3\).

Step 4: Calculating New y-coordinate

To find the new \(y\)-coordinate, add the increment \(\Delta y\) to the initial \(y\)-coordinate: \(y_{new} = 3 - 6 = -3\).

Step 5: Finding the New Position

The new position of the particle after the increments is \((x_{new}, y_{new}) = (3, -3)\).

Key Concepts

Particle Motion

Particle motion in a coordinate plane refers to how a particle moves from one point to another. Imagine a board game, where your piece hops from square to square; with particle motion, it’s like you're moving your token on a giant grid based on certain rules or changes given by coordinates. In the case of the exercise, the particle is initially at the point \((-2, 3)\). When particles move, we usually do so along two axes -- the horizontal (\(x\)-axis) and the vertical (\(y\)-axis). By using specific coordinate increments, which are changes in position, you can determine the new location of the particle. Here, the \(x\)-coordinate increases by 5 and the \(y\)-coordinate decreases by 6. This method of moving particles over a grid is valuable for visualizing different movements in physics or environmental modeling, making coordinate geometry a crucial skill to understand.

Coordinate Increments

Coordinate increments refer to the units you add or subtract from a point's coordinates to find the new position on a grid. In simpler terms, think of increments like steps you take to move to a new spot.In coordinate geometry, increments are given as \(\Delta x\) and \(\Delta y\):

• \(\Delta x\): Change in the horizontal direction, left or right.
• \(\Delta y\): Change in the vertical direction, up or down.

For instance, in our problem, the particle’s new position is calculated by simply adding the increments to its original coordinates:

• New \(x\)-coordinate: \(x_{new} = x + \Delta x = -2 + 5 = 3\)
• New \(y\)-coordinate: \(y_{new} = y + \Delta y = 3 - 6 = -3\)

This systematic approach allows you to predict and verify where the particle ends up after the increments, ensuring you’re always aware of the particle's path and final position.

Two-Dimensional Coordinates

Two-dimensional coordinates are pairs of numbers that define a point's location on a flat plane, like a piece of graph paper. The pair is usually written as \((x, y)\), where:

• \(x\) is the position along the horizontal \(x\)-axis.
• \(y\) is the position along the vertical \(y\)-axis.

Anyone who has played a strategic board game will be familiar with navigating a map in two dimensions. In our initial scenario, the particle starts at point \((-2, 3)\), meaning 2 units left of the origin and 3 units up. After applying the coordinate increments, the particle's position changes to \((3, -3)\), providing a clear example of how two numbers can track movement. This simple numerical system is immensely powerful, forming the basis for both the math behind simple maps we use daily and complex computer graphics in video games. Understanding two-dimensional coordinates is key to grasping more advanced geometric and algebraic concepts.