Problem 21 - Precalculus with Calculus Previews

Question

Find the distance between the given points. $$ (3,-1,2),(6,4,8) $$

Step-by-Step Solution

Answer

The distance between the points is \(\sqrt{70}\).

Step 1: Identify 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, we use the distance formula: \\[d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}\]

Step 2: Substitute the Coordinates

Substitute the given points \(3, -1, 2\) and \(6, 4, 8\) into the formula. This gives us: \\[d = \sqrt{(6 - 3)^2 + (4 - (-1))^2 + (8 - 2)^2}\]

Step 3: Calculate the Differences and Squares

Calculate each difference and then square it: \- \(x_2 - x_1 = 6 - 3 = 3\) so \(3^2 = 9\)- \(y_2 - y_1 = 4 - (-1) = 5\) so \(5^2 = 25\)- \(z_2 - z_1 = 8 - 2 = 6\) so \(6^2 = 36\)

Step 4: Sum the Squares

Add the squares calculated in the previous step: \\[9 + 25 + 36 = 70\]

Step 5: Compute the Square Root

Take the square root of the sum to find the distance: \\[d = \sqrt{70}\] \Thus, the distance between the points is \(\sqrt{70}\).

Key Concepts

Three-Dimensional Distance

Understanding the concept of three-dimensional distance is essential when working with points in space. In a two-dimensional plane, we usually calculate distances with the Pythagorean theorem. However, in three-dimensional space, points have an extra dimension, making the distance calculation slightly more involved.

The formula used to find the distance between two points \(x_1, y_1, z_1\) and \(x_2, y_2, z_2\) incorporates all three axes — x, y, and z. It is expressed as follows:
\[d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}\]
This formula might remind you of the Pythagorean theorem, because it's essentially an extension of it into three dimensions. You calculate the straight-line distance between two points by considering how far you need to travel along each axis. Think of it as determining the hypotenuse of a right triangle formed in 3D space. Understanding this concept will allow you to solve problems involving distances in any spatial context.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, plays a significant role in solving spatial problems involving distances, angles, and lines. It combines algebra and geometry using a coordinate system, like the Cartesian plane, to describe geometric shapes in numerical terms.

In three-dimensional space, coordinate geometry deals with points having three coordinates: x, y, and z. These points are represented in a space defined by three axes. Here's a quick reminder of how each component affects a point:

• The x-coordinate tells you horizontal position.
• The y-coordinate tells you vertical position.
• The z-coordinate adds depth, positioning the point in space.

Using these coordinates, we can measure distances, examine shapes, and find angles. The beauty of coordinate geometry lies in how easily it lets us transition from a geometric problem to an algebraic one by representing geometric figures with equations. It simplifies the problem-solving process by providing a structured method for calculations. Embrace coordinate geometry, as it will deepen your comprehension of spatial problems.

Problem Solving

Problem solving in mathematics often involves breaking down complex tasks into smaller, more manageable steps. Let's examine how this process works when calculating the distance between points in three-dimensional space.

Start by identifying what you know and what you need to find. In our example, you're given two points, and you need to determine the distance between them using the three-dimensional distance formula. Once you have this formula, substitute the given coordinates:
\[d = \sqrt{(6 - 3)^2 + (4 - (-1))^2 + (8 - 2)^2}\]
Next, compute the differences for each coordinate, square them, and sum the squares:

• \((6-3)^2 = 9\)
• \((4-(-1))^2 = 25\)
• \((8-2)^2 = 36\)

Finally, simplify to find the distance:
\[d = \sqrt{70}\]
This structured approach not only simplifies problem solving but also builds essential skills that you can apply to future mathematical challenges. Emphasize practice, as it will make you more proficient at dissecting and addressing various types of problems.