Problem 3 - 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 \(B\) . $$ A(-3.2,-2), B(-8.1,-2) $$

Step-by-Step Solution

Answer

The increments are \(\Delta x = -4.9\) and \(\Delta y = 0\), and the distance is 4.9.

Step 1: Find the Increment in x-coordinates

To find the increment in the x-direction, subtract the x-coordinate of point A from the x-coordinate of point B: \[\Delta x = x_B - x_A = -8.1 - (-3.2)\]Calculate the result to get: \[\Delta x = -8.1 + 3.2 = -4.9\]

Step 2: Find the Increment in y-coordinates

To find the increment in the y-direction, subtract the y-coordinate of point A from the y-coordinate of point B: \[\Delta y = y_B - y_A = -2 - (-2)\]Calculate the result to get: \[\Delta y = 0\]

Step 3: Calculate the Distance from A to B

Use the distance formula to calculate the distance between A and B. The formula is: \[\text{Distance} = \sqrt{(\Delta x)^2 + (\Delta y)^2}\]Substitute the increments found: \[\text{Distance} = \sqrt{(-4.9)^2 + 0^2}\]Calculate the result to get: \[\text{Distance} = \sqrt{24.01} \approx 4.9\]

Key Concepts

Distance Formula

The distance formula is a crucial concept in coordinate geometry that helps us calculate the distance between two points on the coordinate plane. It is derived from the Pythagorean theorem applicable in right-angled triangles.

To find the distance between two points, say \(x_1, y_1\) and \(x_2, y_2\), the formula is:
\[ \text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

This equation yields the straight-line or Euclidean distance between the points. In this exercise, by substituting the increments in coordinates for $$$\Delta x$$ and $$\Delta y$$$, we find:
- \(x_1 = -3.2\) and \(x_2 = -8.1\) lead to \(\Delta x = -8.1 + 3.2 = -4.9\)
- \(y_1 = y_2 = -2\), thus, \(\Delta y = 0\)

Now, applying the distance formula, we find the distance to be:
\[ \text{Distance} = \sqrt{(-4.9)^2 + 0^2} = 4.9 \]

Coordinate Plane

The coordinate plane, often called the Cartesian plane, is a two-dimensional surface where each point is specified by a pair of numerical coordinates. These coordinates are usually written as \(x,y\), where \(x\) denotes the horizontal position (x-axis), and \(y\) denotes the vertical position (y-axis).

Understanding the coordinate plane is fundamental to graphing equations, plotting points, and performing transformations. The plane itself is divided into four quadrants by the intersection of the x-axis and y-axis, which are perpendicular bisectors. Each point’s location is identified by its respective \(x\) and \(y\) values.

In solving problems like these, we map the points from their coordinates. For the exercise, point \(A\) is at \((-3.2, -2)\) and point \(B\) is at \((-8.1, -2)\), making it straightforward to visually or mathematically determine their relative positions in the plane.

Increments in Coordinates

Increments in coordinates represent the change or shift in the x or y values from one point to another within the coordinate plane. This concept is essential to understanding movement or translation between two points.

The increment in the x-direction, \(\Delta x\), is computed by finding the difference between the x-coordinates of the two points:

• \(\Delta x = x_B - x_A\)

The increment in the y-direction, \(\Delta y\), is calculated similarly:

• \(\Delta y = y_B - y_A\)

For the provided exercise:
- \(\Delta x = -8.1 - (-3.2) = -4.9\)
- \(\Delta y = -2 - (-2) = 0\)

This tells us that there is a negative change or a move of 4.9 units to the left along the x-axis, while there is no vertical movement, as shown by the zero value in \(\Delta y\).