Problem 4 - Thomas Calculus

Question

In Exercises \(1-4,\) a particle moves from \(A\) to \(B\) in the coordinate plane. Find the increments \(\Delta x\) and \(\Delta y\) in the particle's coordinates. Also find the distance from \(A\) to \(B\) . $$ A(\sqrt{2}, 4), \quad B(0,1.5) $$

Step-by-Step Solution

Answer

Δx = -√2, Δy = -2.5, distance = √8.25

Step 1: Calculate Δx

Find the difference in the x-coordinates of points A and B. Calculate it using the formula \( \Delta x = x_B - x_A \). Here, \( x_A = \sqrt{2} \) and \( x_B = 0 \). So, \( \Delta x = 0 - \sqrt{2} = -\sqrt{2} \).

Step 2: Calculate Δy

Find the difference in the y-coordinates of points A and B using the formula \( \Delta y = y_B - y_A \). Here, \( y_A = 4 \) and \( y_B = 1.5 \). So, \( \Delta y = 1.5 - 4 = -2.5 \).

Step 3: Calculate the Distance from A to B

To find the distance between points A and B, use the distance formula: \( d = \sqrt{(\Delta x)^2 + (\Delta y)^2} \). Substitute \( \Delta x = -\sqrt{2} \) and \( \Delta y = -2.5 \) into the formula: \( d = \sqrt{(-\sqrt{2})^2 + (-2.5)^2} = \sqrt{2 + 6.25} = \sqrt{8.25} \).

Key Concepts

Particle Motion

When a particle moves in the coordinate plane, it transitions from one point to another. This motion can be expressed in terms of changes in its coordinates. Imagine moving a tiny object from point A to point B on a grid. The path it takes is its **motion**, and we track it by noting how its position changes on the graph.

In coordinate geometry, each point has an "address" given by its x and y coordinates, like a home has a street and a house number. As the particle moves, its journey can be understood by calculating the changes in these coordinates.

• The x-coordinate tells us how far left or right the particle is on the plane.
• The y-coordinate indicates how far up or down it is.

These changes are often described using coordinate increments, which we'll dive into shortly. Understanding the particle's movement in this way allows us to precisely measure how far and in what direction the particle has shifted.

Distance Formula

The distance formula is a crucial tool in coordinate geometry, used to calculate the shortest path between two points on a plane. Imagine you need to travel from point A to point B as quickly and efficiently as possible. That shortest path is the distance.

For two points, A \( (x_A, y_A) \) and B \( (x_B, y_B) \), the distance \( d \) is found using the formula:\[d = \sqrt{(x_B-x_A)^2 + (y_B-y_A)^2}\]By calculating this distance, you find how long a line connecting these two points would be. Keep in mind:

• Squaring the differences ensures that the distance is measured as a positive quantity.
• The square root brings the result back to the original scale of the plane.

Using the distance formula helps you figure out spatial relationships between points, whether you're mapping a journey on a small grid or a vast map.

Coordinate Increments

Coordinate increments refer to the changes in the x and y values of a point as it moves from point A to point B. It's like noting how many steps you take in the east-west or north-south directions when you walk.

To find these changes, use:

• **Δx**: Change in x-coordinate, calculated as \( x_B - x_A \).
• **Δy**: Change in y-coordinate, calculated as \( y_B - y_A \).

These increments help decipher the particle's journey between two points. For instance:- If \( Δx \) is negative, the particle moved to the left.- If \( Δy \) is negative, the particle moved downwards.- Larger absolute values of Δx or Δy indicate farther travel in that direction.Understanding increments is key to mapping out how an object transitions in space, providing clarity on direction and magnitude of its movement.

Coordinate Plane

The coordinate plane is a flat surface defined by a horizontal line (x-axis) and a vertical line (y-axis) intersecting at a point called the origin \( (0, 0) \). This plane acts like a giant piece of graph paper where points can be placed using their coordinates.

Here's a quick breakdown:

• The **x-axis** runs horizontally and tells how far something is from left to right.
• The **y-axis** runs vertically and shows the up and down distance.
• Every point on the plane has a unique pair of coordinates \( (x, y) \).

The coordinate plane is fundamental for visualizing mathematical problems and solutions. It helps in understanding concepts such as slope, lines, and even curves by providing a clear, organized method of plotting data.

Whenever a particle moves in this plane, it's like drawing a small line or dot on a map, helping us see and calculate its trajectory and destination.