Problem 42 - Thomas Calculus

Question

In Exercises \(41-46,\) find the distance between points \(P_{1}\) and \(P_{2}\) $$P_{1}(-1,1,5), \quad P_{2}(2,5,0)$$

Step-by-Step Solution

Answer

The distance between points is \(5\sqrt{2}\).

Step 1: Understand the Distance Formula

To find the distance between two points in 3D space, use the distance formula: \[ \text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2} \] where \(P_1(x_1, y_1, z_1)\) and \(P_2(x_2, y_2, z_2)\) are the coordinates of the two points.

Step 2: Identify Point Coordinates

Identify the coordinates of the given points: \(P_1 = (-1, 1, 5)\) and \(P_2 = (2, 5, 0)\). Here, \(x_1 = -1\), \(y_1 = 1\), \(z_1 = 5\), \(x_2 = 2\), \(y_2 = 5\), and \(z_2 = 0\).

Step 3: Apply the Distance Formula

Substitute the identified coordinates into the distance formula: \[ \text{Distance} = \sqrt{(2 - (-1))^2 + (5 - 1)^2 + (0 - 5)^2} \] Simplify inside the square root to find the distance.

Step 4: Simplify the Expression

Calculate each component separately: \((2 - (-1))^2 = 3^2 = 9\), \((5 - 1)^2 = 4^2 = 16\), and \((0 - 5)^2 = (-5)^2 = 25\). Sum these values: \(9 + 16 + 25 = 50\).

Step 5: Compute the Distance

Take the square root of the sum to find the distance: \[ \text{Distance} = \sqrt{50} \] Simplify to get \[ \text{Distance} = 5\sqrt{2} \].

Key Concepts

Euclidean distance

The Euclidean distance is a fundamental concept in geometry. It's used to measure the straight-line distance between two points in space. In a 3D context, we typically represent these points with three coordinates each:

• Each point has an x (horizontal), y (vertical), and z (depth) coordinate.
• The formula for Euclidean distance between two points \(P_1(x_1, y_1, z_1)\)and \(P_2(x_2, y_2, z_2)\) is \[ \text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2} \]

This formula is derived from the Pythagorean theorem. It considers the difference between corresponding coordinates of the two points as sides of a right triangle. By squaring these differences, we obtain the squares of the lengths of these sides. The sum of these squares gives the square of the Euclidean distance. Remember, the Euclidean distance can never be negative because we use squares and square roots.

Analytic geometry

Analytic geometry, often referred to as coordinate geometry, is the study of geometry using a coordinate system. It uniquely combines algebra and geometry by using coordinates to solve geometric problems easily. This is accomplished by representing geometric shapes like lines, planes, and curves with algebraic equations. By converting geometric problems into algebraic equations:

• We can apply algebraic methods to obtain solutions for geometrical problems.
• This is useful in finding distances, as it allows us not only to work in two dimensions but also to extend our understanding to three dimensions and beyond.

In this framework, points become sets of coordinates, and the distance formula becomes a straightforward application of algebra. Solving problems in analytic geometry involves understanding how changes in one part of a geometric figure (like a point) affect the whole structure. This makes concepts like the Euclidean distance particularly valuable tools.

Coordinate geometry

Coordinate geometry is a branch of mathematics that employs a coordinate plane to describe and solve geometric problems. It's essentially a more practical and applied approach to geometry, where:

• We use a grid-like system (often an x, y, and sometimes z-axis) to define the location of points.
• The relationship between points is analyzed using algebraic concepts.

It allows us to visualize geometric objects by reducing the abstract to concrete coordinates. For example, the 3D distance formula itself is a consequence of this transformation from abstract geometric thinking to the more tangible and visual method of using coordinates. This branch of mathematics provides a way to precisely describe the position and dimensions of shapes and figures, enhancing our ability to solve real-life spatial problems. Thus, coordinate geometry is integral in activities ranging from computer graphics to navigation systems.