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

Step-by-Step Solution

Answer

Δx = -2, Δy = 4, distance = \(2\sqrt{5}\).

Step 1: Determine the Increment in x-coordinate, Δx

To find the change in the x-coordinate (Δx), subtract the x-coordinate of point A from the x-coordinate of point B. Here, the x-coordinate of A is -1 and for B it is -3. So, \( \Delta x = -3 - (-1) = -3 + 1 = -2 \).

Step 2: Determine the Increment in y-coordinate, Δy

To find the change in the y-coordinate (Δy), subtract the y-coordinate of point A from the y-coordinate of point B. Here, the y-coordinate of A is -2 and for B it is 2. So, \( \Delta y = 2 - (-2) = 2 + 2 = 4 \).

Step 3: Calculate the Distance Using the Distance Formula

The distance between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by the distance formula: \( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \). Substituting the given coordinates A(-1, -2) and B(-3, 2) into the formula, we get: \[ d = \sqrt{(-3 - (-1))^2 + (2 - (-2))^2} = \sqrt{(-2)^2 + (4)^2} = \sqrt{4 + 16} = \sqrt{20} = 2\sqrt{5} \].

Key Concepts

Increments in Coordinates: Find the Changes

In coordinate geometry, when a particle moves from one point to another, it experiences changes in its x and y coordinates. We refer to these changes as increments, expressed as \(\Delta x\) and \(\Delta y\). Understanding these increments helps us determine how far the particle has moved along each axis.

To find the increment in the x-coordinate (\(\Delta x\)), you note the starting and ending x-values. Subtract the x-coordinate of the initial point from the x-coordinate of the final point. For instance, if a particle moves from point \(A(-1,-2)\) to point \(B(-3,2)\), the change in the x-coordinate is \(\Delta x = -3 - (-1) = -2\).

Similarly, the increment in the y-coordinate (\(\Delta y\)) is calculated by subtracting the y-coordinate of the initial point from the y-coordinate of the final point. In this example, \(\Delta y = 2 - (-2) = 4\).

• \(\Delta x = x_2 - x_1\)
• \(\Delta y = y_2 - y_1\)

Distance Formula: Measure Exact Travel

Finding how far a particle has traveled in the coordinate plane uses a straightforward mathematical tool called the distance formula. This formula calculates the straight-line distance—often referred to as "as-the-crow-flies" distance—between two points. This understanding is crucial for determining the direct path taken between starting and end points.

The distance formula is: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \] The formula applies the Pythagorean theorem by using the horizontal and vertical increments as two sides of a right triangle, and the distance as the hypotenuse.

By substituting the given values from points \(A(-1, -2)\) and \(B(-3, 2)\):

• Calculate \(\Delta x = -2\) and \(\Delta y = 4\)
• Plug into formula: \(d = \sqrt{(-2)^2 + (4)^2} = \sqrt{20} = 2\sqrt{5}\)

Using this formula, we determine that the distance from \(A\) to \(B\) is \(2\sqrt{5}\). This represents the length of the direct path between the two points.

Particle Movement in Plane: Visualizing the Path

When examining particle movement within the coordinate plane, we focus on how the particle transitions from one point to another. The concept ties into both the increments and the distance formula by outlining the change's trajectory.

Picture the coordinate plane as a grid where each point is a location for the particle at a given time. The particle's movement involves a clear change in position along two axes:

• The x-axis, representing left or right movement.
• The y-axis, representing up or down movement.

Visualizing the move from \(A(-1, -2)\) to \(B(-3, 2)\), you can draw a line segment connecting the two. The increments \(\Delta x = -2\) and \(\Delta y = 4\) reflect a move left by 2 units and up by 4 units. Combined, this movement forms a diagonal path that illustrates the direct journey between these two coordinate points.