Problem 6 - Calculus with Concepts in Calculus

Question

Determine the distance between the given points. \((-1,-3)\) and \((-2,2)\)

Step-by-Step Solution

Answer

The distance between the points is \(\sqrt{26}\).

Step 1: Understand the Distance Formula

The distance between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) in a coordinate plane can be calculated using the distance formula: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \] where \( d \) is the distance between the points.

Step 2: Identify Coordinates

Here, the given points are \( (-1, -3) \) and \( (-2, 2) \). For these points, assign the coordinates: \( x_1 = -1 \), \( y_1 = -3 \), \( x_2 = -2 \), and \( y_2 = 2 \).

Step 3: Substitute into the Distance Formula

Substitute the identified coordinates into the distance formula: \(d = \sqrt{(-2 - (-1))^2 + (2 - (-3))^2} \) which simplifies to: \( \sqrt{(-2 + 1)^2 + (2 + 3)^2} \).

Step 4: Simplify the Expression

Simplify inside the square root: \( \sqrt{(-1)^2 + (5)^2} \).Calculate the squares: \( \sqrt{1 + 25} \).

Step 5: Calculate the Distance

Continue simplifying: \( \sqrt{26} \).The distance between the points is \( \sqrt{26} \).

Key Concepts

Understanding the Coordinate Plane

Imagine a grid that spans in four directions: right, left, up, and down. This is called the coordinate plane. It helps us locate points using two numbers, which we call coordinates. The first number, known as the x-coordinate, tells us how far to move left or right. The second number, known as the y-coordinate, shows how far to move up or down. For example, the point \((-1, -3)\) means you move one step left (as \(-1\) is negative) and three steps down (since \(-3\) is also negative). Similarly, the point \((-2, 2)\) means two steps left and two steps up. The beauty of the coordinate plane is that it helps visualize and solve problems involving distances between points using simple calculations.

Steps in Calculating Distance

Calculating the distance between two points using the distance formula involves several simple steps. Let's break them down:

• Step 1: Understand the formula - It's essential to know the distance formula, which is \(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\). This formula comes from the Pythagorean Theorem, and helps calculate how far apart two points are on the coordinate plane.
• Step 2: Identify the coordinates - Look at your points: \((-1, -3)\) and \((-2, 2)\). Here, you have \(x_1 = -1\), \(y_1 = -3\), \(x_2 = -2\), and \(y_2 = 2\).
• Step 3: Substitute into the formula - Simply insert the coordinates into the formula making sure to follow order operations: \(d = \sqrt{(-2 - (-1))^2 + (2 - (-3))^2}\).
• Step 4: Simplify the expression - Work out the operations inside the square root: \(d = \sqrt{(-1)^2 + (5)^2}\).
• Step 5: Calculate the distance - Finish by solving to get \(d = \sqrt{26}\). This gives the distance between the points.

Breaking up the process like this makes it easier to handle each part without getting overwhelmed. It's a step-by-step path to find exact distances between points on a grid.

Finding Distance Between Points

When two points are plotted on a coordinate plane, calculating the distance between them involves using the Pythagorean Theorem, embedded in the distance formula. The formula shows the relationship between the x and y differences of the points:

• The change in x (known as "horizontal change") is calculated as \((x_2 - x_1)\).
• The change in y ("vertical change") is calculated as \((y_2 - y_1)\).

These changes form the legs of a right triangle, where the distance we are trying to find is the hypotenuse. By substituting and squaring these differences, we can find the distance without drawing the triangle.For our given points \((-1, -3)\) and \((-2, 2)\), we calculated the distance to be \(\sqrt{26}\). This result shows the straight-line distance, or the shortest path, between these two points. Remember, understanding how changes in coordinates relate to distance is a core concept in coordinate geometry.