Problem 38 - Thomas Calculus

Question

A particle starts at \(A(6,0)\) and its coordinates change by increments \(\Delta x=-6, \Delta y=0 .\) Find its new position.

Step-by-Step Solution

Answer

The particle's new position is \((0,0)\).

Step 1: Understand the Given Coordinates

The particle initially starts at point \(A(6,0)\). This means the starting \(x\)-coordinate is 6 and the starting \(y\)-coordinate is 0.

Step 2: Identify the Changes in Coordinates

The increment for the \(x\)-coordinate is given as \(\Delta x = -6\), which means that 6 will be subtracted from the \(x\)-coordinate. The increment for the \(y\)-coordinate is \(\Delta y = 0\), indicating there is no change in the \(y\)-coordinate.

Step 3: Apply the Changes to the Coordinates

Subtract \(\Delta x\) from the initial \(x\)-coordinate: \(6 - 6 = 0\). Keep the \(y\)-coordinate unchanged since \(\Delta y = 0\). Therefore, the new \(y\)-coordinate remains 0.

Step 4: Determine the New Position

Combine the results from the previous step to determine the new position of the particle. The new coordinates are \((0,0)\).

Key Concepts

Particle Motion

In coordinate geometry, understanding particle motion involves tracking how a point, or particle, moves on a plane following specified changes. This concept is fundamental in various fields such as physics, engineering, and computer graphics.

At the start, a particle's position is represented by coordinates, such as the point \(A(6,0)\) in our example. As it moves, the coordinates change based on given increments or decrements. These changes can be expressed as \(\Delta x\) for the horizontal shift and \(\Delta y\) for the vertical shift.

When studying particle motion, it's crucial to grasp how these incremental changes affect the overall position, providing insights into directionality and path taken by the particle. For instance, a negative \(\Delta x\) indicates a move leftward, while a \(\Delta y\) of zero reflects no vertical shift.

Coordinate Changes

In the context of coordinate geometry, coordinate changes refer to the adjustments made to a point's position based on specific increments or decrements.

These changes are denoted by \(\Delta x\) and \(\Delta y\), which indicate alterations in the x-axis and y-axis respectively.

• Positive \(\Delta x\) values imply a move to the right.
• Negative \(\Delta x\) values imply a move to the left, as seen in our example with \(\Delta x = -6\).
• Positive \(\Delta y\) values imply a move upward.
• Negative \(\Delta y\) values imply a move downward.

Changes allow a straightforward way to shift points and are essential in understanding shifts and transformations in a two-dimensional space. For example, applying a \(\Delta x\) of \-6\ resulted in moving the particle from \((6, 0)\) to \((0, 0)\).

Position Calculation

Position calculation is the process of determining the new position of a particle after specified coordinate changes.

To calculate this, simply apply the changes using the formulas:

• New \(x\)-coordinate: \(x_{new} = x_{initial} + \Delta x\)
• New \(y\)-coordinate: \(y_{new} = y_{initial} + \Delta y\)

For instance, starting from \(A(6,0)\), and applying \(\Delta x = -6\) and \(\Delta y = 0\) results in:

• New \(x\)-coordinate: \(6 - 6 = 0\)
• New \(y\)-coordinate: \(0 + 0 = 0\)

Hence, the new position is \((0,0)\). This calculation is crucial in determining the exact location of a point after motion and is foundational for analyzing movements within coordinate systems.