Problem 31 - Precalculus with Calculus Previews

Question

Use the distance formula to determine whether the given points are collinear. $$ P_{1}(1,0,4), P_{2}(-4,-3,5), P_{3}(-7,-4,8) $$

Step-by-Step Solution

Answer

The points are not collinear.

Step 1: Understand the Problem

We need to determine if the given points \(P_1(1,0,4), P_2(-4,-3,5), P_3(-7,-4,8)\) are collinear. To do this, we will use the distance formula to find the lengths of the segments connecting these points.

Step 2: Recall the Distance Formula

The distance formula for two points \((x_1, y_1, z_1)\) and \((x_2, y_2, z_2)\) in 3D space is: \(\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}.\)

Step 3: Calculate Distance \(P_1P_2\)

Using the distance formula, calculate \(P_1P_2\):\[\sqrt{((-4) - 1)^2 + ((-3) - 0)^2 + (5 - 4)^2} = \sqrt{(-5)^2 + (-3)^2 + (1)^2} = \sqrt{25 + 9 + 1} = \sqrt{35}.\]

Step 4: Calculate Distance \(P_2P_3\)

Next, find the distance \(P_2P_3\):\[\sqrt{((-7) - (-4))^2 + ((-4) - (-3))^2 + (8 - 5)^2} = \sqrt{(-3)^2 + (-1)^2 + (3)^2} = \sqrt{9 + 1 + 9} = \sqrt{19}.\]

Step 5: Calculate Distance \(P_1P_3\)

Finally, calculate \(P_1P_3\):\[\sqrt{((-7) - 1)^2 + ((-4) - 0)^2 + (8 - 4)^2} = \sqrt{(-8)^2 + (-4)^2 + (4)^2} = \sqrt{64 + 16 + 16} = \sqrt{96}.\]

Step 6: Analyze the Results

If the points are collinear, the sum of the two smaller distances should equal the largest distance. Calculate \(\sqrt{35} + \sqrt{19}\) and compare to \(\sqrt{96}\).Since \(\sqrt{35} + \sqrt{19} eq \sqrt{96}\), the points are not collinear.

Key Concepts

Collinear Points

Three points are described as collinear if they all lie on a single straight line. In mathematics, determining whether points in space are collinear can be crucial in analyzing their geometric properties and relationships.

To check if points are collinear, one common method involves checking the distances between them. For three given points, you can calculate the three distances that connect the points. After calculating these distances, we analyze whether the sum of the lengths of the two shorter segments equals the length of the longer segment.

Here's a quick checklist to remember:

• Calculate the distances between all pairs of points.
• Identify the longest of these distances.
• Check if the sum of the other two distances equals the longest distance.

If the lengths satisfy this condition, then the points are collinear. If not, the points are not collinear, meaning they do not form a straight line but instead form distinct, non-linear triangles or shapes in the space.

3D Space

3D space, short for three-dimensional space, is a mathematical concept where space has three dimensions: length, width, and height (or depth). It is the environment where all real physical forms exist and interact. When we refer to points, lines, or any geometrical concepts in 3D space, we are considering their positions and orientations in this volumetric environment.

In 3D space, points are defined by three coordinates, usually represented as \(x, y, z\). These coordinates specify a point's exact location along each respective axis. When performing calculations in three-dimensional geometry, you often need to consider relationships and distances between points, planes, and lines in these three dimensions.

When working with 3D space, remember:

• Use three coordinates for points \(x, y, z\).
• Understand that visualizing spatial relationships can aid in solving geometric problems.
• Be comfortable using the distance formula in this extended space to measure distances between points.

Grasping the concept of 3D space can initially be challenging because it requires thinking beyond the two-dimensional plane we see on paper. However, once understood, it beautifully opens up a whole new realm of geometry.

Distance Calculation

Calculating the distance between two points in 3D space is fundamental in geometry. The distance formula allows us to find this distance by extending the basic Pythagorean theorem. It helps us compute the shortest path between these points in the environment. Given two points, \( (x_1, y_1, z_1)\) and \( (x_2, y_2, z_2)\), the distance formula in three-dimensional space is expressed as: \[ \sqrt{ (x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2 } \]This formula efficiently applies the concept of three-dimensional squares for each coordinate pair difference and links them to provide a single maximum length.

If you are looking to solve a problem using this formula:

• Substitute your point coordinates into the formula.
• Calculate each squared difference for all three dimensions.
• Add these squared values together.
• Take the square root of the sum to find the distance between the points.

The application of the distance formula is straightforward but requires particular care in following operations like squaring differences correctly and ensuring calculations are accurate. This simple tool becomes incredibly powerful whether you're calculating distances in physics problems, engineering tasks, or any field needing 3D spatial analysis.