Problem 10 - Calculus with Concepts in Calculus

Question

Determine the distance between the given points. \((a, b)\) and \((b, a)\)

Step-by-Step Solution

Answer

The distance is \\( \sqrt{2} \times |b-a| \\).

Step 1: Recognize the Distance Formula

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

Step 2: Identify the Coordinates

Identify the coordinates for the points \( (a, b) \) and \( (b, a) \). Here, \( x_1 = a, y_1 = b \) and \( x_2 = b, y_2 = a \).

Step 3: Substitute the Coordinates into the Distance Formula

Substitute the identified coordinates into the distance formula: \[ d = \sqrt{(b - a)^2 + (a - b)^2} \]

Step 4: Simplify the Expression

Notice that \( (b-a)^2 = (a-b)^2 \), since squaring negates signs. Therefore, simplify:\[ d = \sqrt{2(b-a)^2} \] Further simplify: \d = \sqrt{2} \times \lvert b-a \rvert \.

Step 5: Final Calculation

The final distance can be represented as:\[ d = \sqrt{2} \times |b-a| \]

Key Concepts

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is a branch of mathematics that uses a coordinate system to describe geometric shapes. It involves using coordinates on a plane to define the position of points and the relationships between them. In coordinate geometry, any point can be defined using an ordered pair of numbers, usually written as \(x, y\). These coordinates tell us where the point lies on the Cartesian plane.

When working with coordinate geometry, we can explore and solve various geometric problems by using algebraic methods. This field combines geometry and algebra, providing a way to study the properties and relationships of geometric figures through equations. It utilizes the x-axis and y-axis to plot points and visualize distances, lines, and shapes. This is especially useful for calculating distances between points, such as in the distance problem you're working on.

Distance Between Points

Calculating the distance between two points in a coordinate plane involves understanding and applying a specific formula. The distance formula is derived from the Pythagorean Theorem and it helps to find the length of the line segment connecting two points in a coordinate system.

For any two points, say \(x_1, y_1\) and \(x_2, y_2\), the distance \(d\) can be 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 between the x and y coordinates of the two points. By calculating, you measure how far apart the two points are directly on a coordinate plane.
In your problem, the points are \(a, b\) and \(b, a\). Substitute these into the formula to find the distance, which simplifies variously through understanding symmetry and properties of the squaring operation.

Square Root Simplification

Simplifying square roots involves finding the simplest form of a square root expression. During the distance calculation, you may encounter the need to simplify the square root, which makes it easier to interpret or calculate the answer. This process involves identifying perfect squares within the expression and simplifying them.

Taking the expression \(\sqrt{2(b-a)^2}\), the simplification occurs by recognizing that the expression inside the square root simplifies. Since \((b-a)^2\) means squaring the difference, it doesn't matter whether it's \(b-a\) or \(a-b\) because squaring makes it positive. Thus, \((b-a)^2 \) and \(2(b-a)^2\) helps extract the square root partly which gives:

• \(\sqrt{2} \times |b-a|\)

This expression is more concise and represents the distance using the absolute difference multiplied by the square root of 2, a suitable representation for geometrical distance.