Problem 1 - 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 \(B\) . $$ A(-3,2), \quad B(-1,-2) $$

Step-by-Step Solution

Answer

\(\Delta x = 2\), \(\Delta y = -4\), and the distance is \(2\sqrt{5}\).

Step 1: Calculate \( \Delta x \)

To calculate the change in the x-coordinate, subtract the x-coordinate of point A from the x-coordinate of point B. Thus, \( \Delta x = x_B - x_A = -1 - (-3) = 2 \).

Step 2: Calculate \( \Delta y \)

To calculate the change in the y-coordinate, subtract the y-coordinate of point A from the y-coordinate of point B. Thus, \( \Delta y = y_B - y_A = -2 - 2 = -4 \).

Step 3: Determine the Distance from A to B

Apply the distance formula: \[ d = \sqrt{(x_B - x_A)^2 + (y_B - y_A)^2} \] Substituting the values: \[ d = \sqrt{(-1 - (-3))^2 + (-2 - 2)^2} = \sqrt{2^2 + (-4)^2} = \sqrt{4 + 16} = \sqrt{20} = 2\sqrt{5} \].

Key Concepts

Distance Formula

The distance formula is a useful tool in coordinate geometry employed to measure the distance between two points on a coordinate plane. It is derived from the Pythagorean Theorem and helps us find the straight-line distance between two locations given their coordinates. For two points, \((x_1, y_1)\) and \((x_2, y_2)\) in the coordinate plane, the distance \(d\) is calculated using the formula: \[d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\] This formula essentially measures the hypotenuse of a right triangle formed by the horizontal and vertical distances (or increments) between the points. This gives us an accurate measure of the true distance between two points, regardless of their position on the plane. To apply the formula successfully, it’s crucial to ensure that the correct coordinates are substituted into the equation.

Increments in Coordinates

When we discuss increments in coordinates, we refer to the changes or differences in the x and y values as we move from one point to another on the coordinate plane. This is often represented as \(\Delta x\) and \(\Delta y\).

• \(\Delta x = x_{final} - x_{initial}\)
• \(\Delta y = y_{final} - y_{initial}\)

These increments show how much the x-values and y-values change from one point to another. Understanding these increments is fundamental in determining the slope or direction of the line when a particle or point moves from position \(A\) to position \(B\). It is crucial in various applications, such as physics, to analyze movement or trajectory.

Coordinate Plane

The coordinate plane is a two-dimensional surface where we can plot points, lines, and curves. It is defined by two perpendicular axes: the x-axis (horizontal) and the y-axis (vertical). Points on this plane are represented by ordered pairs \((x, y)\), which indicate their position in relation to these axes. The origin, where both axes intersect, is the point \((0, 0)\). Positive x-values lie to the right, and negative x-values lie to the left of the origin. Similarly, positive y-values are above, and negative y-values are below the origin. Exploring the coordinate plane helps in visualizing and solving many mathematical problems, including finding distances, midpoints, and analyzing relationships between different points and lines.

Mathematics Problem Solving

Mathematics problem solving is a process that involves understanding a problem, planning a solution, carrying out the plan, and evaluating the results. You often find that real-world scenarios, like tracking a moving particle in a coordinate plane, require multiple steps to solve. Here's how you can approach such a problem effectively:

• **Understand the Problem:** Clarify what you need to find. For example, determine increments and distance between two points.
• **Plan the Solution:** Decide on the mathematical formulas or concepts needed, like the distance formula or calculating increments.
• **Carry Out the Plan:** Substitute the correct values into your chosen formulas and calculate carefully.
• **Review the Results:** Check if your results make sense in the context of the problem and if the steps were correctly followed.

This strategic approach not only helps in obtaining the correct answer but also aids in developing strong analytical and logical thinking skills, essential in mathematics.