Problem 30 - University Calculus: Early Transcendentals

Question

Find the distance between points \(P_{1}\) and \(P_{2}\) $$P_{1}(5,3,-2), \quad P_{2}(0,0,0)$$

Step-by-Step Solution

Answer

The distance between points \(P_1\) and \(P_2\) is \(\sqrt{38}\) or approximately 6.16.

Step 1: Identify the Coordinates

First, let's identify the coordinates of the points given. We have point \(P_1\) with coordinates \((5, 3, -2)\) and point \(P_2\) with coordinates \((0, 0, 0)\).

Step 2: Use the Distance Formula in 3D

The formula to find the distance between two points \((x_1, y_1, z_1)\) and \((x_2, y_2, z_2)\) in three-dimensional space is: \[d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}\]

Step 3: Substitute the Coordinates into the Formula

Substitute the coordinates \(x_1 = 5\), \(y_1 = 3\), \(z_1 = -2\), and \(x_2 = 0\), \(y_2 = 0\), \(z_2 = 0\) into the distance formula:\[d = \sqrt{(0 - 5)^2 + (0 - 3)^2 + (0 + 2)^2}\]

Step 4: Calculate the Squares

Calculate the squares for each coordinate difference:- \((0 - 5)^2 = 25\)- \((0 - 3)^2 = 9\)- \((0 + 2)^2 = 4\)

Step 5: Sum the Squares

Add up the squares calculated in the previous step:\[25 + 9 + 4 = 38\]

Step 6: Take the Square Root

Finally, take the square root of the sum to find the distance:\[d = \sqrt{38}\]This is approximately equal to 6.16.

Key Concepts

Coordinate Geometry

Coordinate geometry is a branch of mathematics that helps us describe and understand the space around us using numbers and formulas. It involves placing points in a coordinate plane and using algebra to solve geometric problems. In a two-dimensional space, we use an

• x-axis
• y-axis

On a three-dimensional space, we add a

• z-axis

Each point in this system is defined by a triplet of numbers, each representing a coordinate along these axes. For example, the point

• \((5, 3, -2)\)

represents a location that is 5 units along the x-axis, 3 units along the y-axis, and -2 units along the z-axis. Understanding coordinate geometry is essential for tackling many distance and space problems, as it allows for precise calculations and clear visualizations.

Distance Calculation

Distance calculation in coordinate geometry involves finding the exact length between two points. Whether in two or three dimensions, this is possible through the distance formula. The formula in three-dimensional space is rooted in the Pythagorean theorem but expanded. To find the distance between

• two points \( (x_1, y_1, z_1) \)
• \( (x_2, y_2, z_2) \)

we use:\[d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}\]Start by finding the difference in each coordinate, square these differences, then add them together. Finally, take the square root of this sum to get the distance. Practicing this process helps build a strong foundational understanding of spatial relationships in mathematics.

Three-Dimensional Space

Three-dimensional space is the environment where every object has length, width, and height. It is the real world we live in, which makes understanding it crucial for various applications, such as in physics and engineering. In 3D space, each point is determined using three coordinates

• (x, y, z)

The z-axis adds depth to the plane formed by the x and y axes, offering a sense of thickness or volume. By utilizing the rules of three-dimensional geometry, like the 3D Distance Formula, one can precisely measure gaps, boundaries, and distances between objects or points. Grasping three-dimensional concepts allows us to see beyond flat surfaces, imagining and crafting ideas that are multi-layered and complex.