Problem 15 - Calculus an Applied Approach

Question

Find the distance between the two points. $$ (-1,-5,7),(-3,4,-4) $$

Step-by-Step Solution

Answer

The distance between the two points is \( \sqrt{206} \) units.

Step 1: Identify the coordinates

First, we identify the coordinates of our two points. For point 1, we have \((x_1, y_1, z_1) = (-1, -5, 7)\) and for point 2, we have \((x_2, y_2, z_2) = (-3, 4, -4)\).

Step 2: Apply the distance formula

Next, we plugging these values into our distance formula. \[ d = \sqrt{{(-3 - (-1))}^2 + {(4 - (-5))}^2 + {(-4 - 7)}^2} \]

Step 3: Do the calculations

Carry out the calculations in the square root. \[ d = \sqrt{{(-2)}^2 + {9}^2 + {-11}^2} \] \[ d = \sqrt{4 + 81 + 121} \] \[ d = \sqrt{206} \]

Key Concepts

Distance Between Points

When trying to measure how far apart two points are in space, you use the 3D distance formula. This formula is an extension of the Pythagorean Theorem and is used to find the distance between points in a three-dimensional environment. It’s much like finding the length of a hypotenuse in a right triangle, only in 3D spaces, we consider all three axis: x, y, and z.
Imagine two points floating in space. Each point has an address determined by coordinates \((x, y, z)\). To find out how far they are from each other, you calculate the difference in each of these three directions.

• Calculate how far apart the points are, along the x-axis.
• Do the same for both the y-axis and the z-axis.

The distance formula stitches these differences together using squares and square roots, producing a single number representing the distance.
Analytically, it reflects how you would visually measure the straight line connecting those two points in space. The resulting value is always a positive number because distance can't be negative.

Coordinate Geometry

Coordinate geometry is all about representing geometrical shapes and figures using numbers and equations. It gives us the tools to describe lines, curves, and shapes in a mathematical framework. When dealing with points in 3D, as in our exercise, the coordinates \((x, y, z)\) help to pinpoint exact locations in space.
Think of a 3D Cartesian plane, where each of the three axes (x, y, z) allows you to specify a point uniquely with three numbers. These coordinates tell you everything about that point's position relative to a set reference point, usually the origin \((0, 0, 0)\).
Coordinate geometry provides us with essential tools, such as:

• The distance formula, which calculates the length between two such points.
• Equations of planes, which can separate space into regions.

These tools become valuable when solving complex problems involving shapes, distances, and intersections in three-dimensional spaces.

Distance Calculation

To actually compute the distance using coordinates, you follow a simple series of steps:
First, take the differences of the corresponding coordinates of the two points. For instance, if you have two points \((x_1, y_1, z_1)\) and \((x_2, y_2, z_2)\), calculate \((x_2 - x_1)\), \((y_2 - y_1)\), and \((z_2 - z_1)\).
For our exercise’s coordinates \((-1, -5, 7)\) and \((-3, 4, -4)\):

• Along the x-axis: Difference is \(-3 - (-1) = -2\)
• Y-axis: \(4 - (-5) = 9\)
• Z-axis: \(-4 - 7 = -11\)

Next, square each of these difference values.

• \((-2)^2 = 4\)
• \(9^2 = 81\)
• \((-11)^2 = 121\)

Finally, add up all the squared numbers and take the square root of the sum to find the distance: \[ d = \sqrt{4 + 81 + 121} = \sqrt{206} \]
This procedure neatly ties together numerical calculations with geometrical concepts of space, revealing the length of the direct line segment linking the two points.