Problem 2 - Calculus Early Transcendentals: Pearson New International Edition

Question

In Problems \(1-4\), plot the given points in the coordinate plane and then find the distance between them. \((-3,5),(2,-2)\)

Step-by-Step Solution

Answer

The distance between the points \((-3,5)\) and \((2,-2)\) is \(\sqrt{74}\).

Step 1: Understand the Coordinates

The problem provides two points, \((-3,5)\) and \((2,-2)\). These points are in the form \((x_1, y_1)\) and \((x_2, y_2)\) where \(x_1 = -3\), \(y_1 = 5\), \(x_2 = 2\), and \(y_2 = -2\). In order to plot the points, we identify the x-coordinate as the horizontal placement and the y-coordinate as the vertical placement.

Step 2: Plot the Points

Draw a coordinate plane with horizontal and vertical axes. Locate the first point at the position where x is -3 and y is 5, and mark it. Similarly, locate the second point at x = 2 and y = -2 and mark it. These marks represent the points \((-3,5)\) and \((2,-2)\) on the graph.

Step 3: Apply the Distance Formula

The distance \(d\) between two points \((x_1, y_1)\) and \((x_2, y_2)\) in a coordinate plane is found using the formula:\[d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]Substituting the given values into the formula:\[d = \sqrt{(2 - (-3))^2 + (-2 - 5)^2}\]

Step 4: Calculate the Difference for Each Coordinate

Find the difference in the x-coordinates: \(x_2 - x_1 = 2 - (-3) = 2 + 3 = 5\). Find the difference in the y-coordinates: \(y_2 - y_1 = -2 - 5 = -7\).

Step 5: Find the Squares of the Differences

Calculate the square of each difference:\((x_2 - x_1)^2 = 5^2 = 25\)\((y_2 - y_1)^2 = (-7)^2 = 49\)

Step 6: Find the Sum of the Squares

Add the squares of the differences:\(25 + 49 = 74\).

Step 7: Calculate the Distance

Take the square root of the sum of the squares to find the distance:\(d = \sqrt{74}\).

Key Concepts

Distance Formula

The distance formula is a key concept in coordinate geometry. It helps us find the length of a line segment between two points on the coordinate plane. In general, the formula is given by:

• \(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\)

This formula is derived from the Pythagorean theorem, which relates the sides of a right triangle. Here, the horizontal side or width is \(x_2 - x_1\), and the vertical side or height is \(y_2 - y_1\). By applying this, you can convert finding distance on the plane to solving for the hypotenuse of a right triangle. Using this straightforward formula, you can calculate the distance between any two given points easily and reliably.

Coordinate Plane

A coordinate plane is a two-dimensional surface where we can plot points defined by two numbers: an x-coordinate and a y-coordinate. It is initiated by two perpendicular lines called axes:

• The horizontal line is the x-axis.
• The vertical line is the y-axis.

The point where these two axes intersect is called the origin, typically denoted as \((0,0)\). This system allows us to specify the exact location of any point in the plane by using these two measurements. As the problem suggests, points such as (-3, 5) and (2, -2) can be located precisely by using their x and y values in the plane.

Plotting Points

Plotting points involves locating and marking specific positions on the coordinate plane based on given coordinates. Here’s how you can plot a point efficiently:

• Start by identifying the x-coordinate, which tells you how far to move horizontally from the origin. A positive x moves right, whereas a negative x moves left.
• Next, identify the y-coordinate, indicating your vertical movement from the origin: upwards if positive, downwards if negative.
• For example, for the point (-3, 5), you first move 3 units left from the origin on the x-axis, then 5 units up on the y-axis.
• Similarly, for the point (2, -2), move 2 units right and then 2 units down.

By marking these positions, you plot the points accurately, enabling you to visualize and calculate further properties, like the distance between them.

Calculating Distance

Calculating distance using coordinates involves several steps that require careful arithmetic to ensure accuracy. Here’s a simple breakdown of the process:

• First, identify the differences in the corresponding coordinates of the two points: calculate \(x_2 - x_1\) and \(y_2 - y_1\).
• Then, square each of these differences to eliminate any negative signs and work with positive values: \((x_2 - x_1)^2\) and \((y_2 - y_1)^2\).
• Add the squared values to determine the sum of the squares: this represents the square of the distance.
• Finally, take the square root of the sum you calculated. This step finds the actual distance, giving you a non-negative number that represents the shortest path between the two points.

Using these systematic steps, you ensure an accurate distance measurement between points on any coordinate plane.