Problem 6 - Algebra and Trigonometry

Question

\(3-6\) Two points \(P\) and \(Q\) are given. (a) Plot \(P\) and \(Q .(b)\) Find the distance between \(P\) and \(Q\) $$ P(5,-4,-6), Q(8,-7,4) $$

Step-by-Step Solution

Answer

The distance between the points is approximately 10.86 units.

Step 1: Understanding the Task

We are given two points, \(P(5,-4,-6)\) and \(Q(8,-7,4)\), and our task is to first plot these points and then calculate the distance between them.

Step 1: Plotting the Points

First, we need to plot the points \(P\) and \(Q\) in a 3D coordinate system. Point \(P\) is located at \(x=5\), \(y=-4\), and \(z=-6\). Point \(Q\) is at \(x=8\), \(y=-7\), \(z=4\). In a 3D plot, each point has three coordinates, and you can plot them accordingly in a graphing software or on paper.

Step 2: Formula for Distance Between Two Points 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 3D is: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}\]

Step 3: Substituting Values into the Formula

Now, substitute the coordinates of points \(P\) and \(Q\) into the formula: \[ d = \sqrt{(8 - 5)^2 + (-7 + 4)^2 + (4 + 6)^2}\]

Step 4: Calculating the Distance

Simplify and calculate the expression:1. Calculate the differences: \( (8-5)=3 \), \( (-7+4)=-3 \), \( (4+6)=10 \)2. Square each difference: \( 3^2=9 \), \( (-3)^2=9 \), \( 10^2=100 \)3. Add the squares: \( 9 + 9 + 100 = 118 \)4. Take the square root of the sum: \( \sqrt{118} \approx 10.86 \)

Key Concepts

Distance Between Two Points

When it comes to understanding the distance between two points in three-dimensional space, we use the 3D distance formula. This formula allows us to calculate the straight-line distance between any two points that have coordinates in the form \( (x, y, z) \). It's like stretching a string between the points in space and measuring its length.To grasp it better, imagine you need to find the distance between points \( P(5, -4, -6) \) and \( Q(8, -7, 4) \). The formula used is:\[d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}\]This formula is an extension of the Pythagorean theorem into three dimensions. By inserting the coordinates for \( P \) and \( Q \), we calculate the differences for each pair \( (x, y, z) \), square them, sum them, and then take the square root. The calculation steps lead us to find the distance \( d \approx 10.86 \), which tells us how far the two points are in space.

Plotting Points in 3D

Plotting points in a 3D coordinate system can be very insightful. It helps visualize spatial relationships and understand abstract math concepts.To plot a point like \( P(5, -4, -6) \), you think of three axes: the x-axis, y-axis, and z-axis.

• The x-axis runs horizontally.
• The y-axis runs vertically on a paper, but in space, imagine it comes out and goes back into the paper as well.
• The z-axis slices through, adding depth.

To find point \( P \), you move 5 units along the x-axis, 4 units down on the y-axis, and then 6 units below the level of the x-y plane (the z-axis indicates depth with negative values going downwards).Point \( Q(8, -7, 4) \), on the other hand, requires moving 8 units along x, 7 down y, and 4 units upwards in the z direction, providing a good exercise in visualizing depth. Using graphing tools can help make these visualizations cleaner and more accurate.

3D Coordinate System

The 3D coordinate system expands on the familiar 2D plane by adding a third axis, enhancing our ability to model and analyze real-world scenarios like architecture or physics. In three dimensions:

• The x-axis remains your guideline for left and right movement.
• The y-axis indicates up and down, but in conjunction with depth when visualized.
• The z-axis adds the 'in-and-out' dimension, signifying depth.

All coordinates in 3D follow the format \( (x, y, z) \), with z being the new component. This complex system enables us to precisely locate points and analyze distances or directions. You imagine these axes as intersecting perpendicularly to form a 3D grid. Starting from the origin (0, 0, 0), any point in this space can be reached by moving the respective number of units along each axis. This understanding is crucial for advanced mathematics and related fields, offering a practical framework to handle complex spatial data.