Problem 29 - Advanced Engineering Mathematics

Question

Use the distance formula to prove that the given points are collinear. $$ P_{1}(1,2,0), P_{2}(-2,-2,-3), P_{3}(7,10,6) $$

Step-by-Step Solution

Answer

The points are not collinear as the distance conditions for collinearity are not satisfied.

Step 1: Understand Collinearity

For points to be collinear, they must lie on the same straight line. We can prove this by checking if the sum of the distances between two pairs of points equals the distance between the farthest points among the three, effectively showing they lie on the same straight path.

Step 2: Recall the Distance Formula

The 3D distance between two points \( (x_1, y_1, z_1) \) and \( (x_2, y_2, z_2) \) is given by: \[ d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2 + (z_2-z_1)^2} \] This formula will be used to calculate the distances between the pairs of points.

Step 3: Calculate Distance \( d_{12} \) Between \( P_{1} \) and \( P_{2} \)

Using the given points \( P_{1}(1,2,0) \) and \( P_{2}(-2,-2,-3) \): \[ d_{12} = \sqrt{((-2) - 1)^2 + ((-2) - 2)^2 + ((-3) - 0)^2} \] \[ d_{12} = \sqrt{(-3)^2 + (-4)^2 + (-3)^2} \] \[ d_{12} = \sqrt{9 + 16 + 9} = \sqrt{34} \]

Step 4: Calculate Distance \( d_{23} \) Between \( P_{2} \) and \( P_{3} \)

Using the given points \( P_{2}(-2,-2,-3) \) and \( P_{3}(7,10,6) \): \[ d_{23} = \sqrt{(7 - (-2))^2 + (10 - (-2))^2 + (6 - (-3))^2} \] \[ d_{23} = \sqrt{(9)^2 + (12)^2 + (9)^2} \] \[ d_{23} = \sqrt{81 + 144 + 81} = \sqrt{306} \]

Step 5: Calculate Distance \( d_{13} \) Between \( P_{1} \) and \( P_{3} \)

Using the given points \( P_{1}(1,2,0) \) and \( P_{3}(7,10,6) \): \[ d_{13} = \sqrt{(7 - 1)^2 + (10 - 2)^2 + (6 - 0)^2} \] \[ d_{13} = \sqrt{(6)^2 + (8)^2 + (6)^2} \] \[ d_{13} = \sqrt{36 + 64 + 36} = \sqrt{136} \]

Step 6: Verify the Collinearity Condition

For the points to be collinear, the sum of the two shorter distances \( d_{12} \) and \( d_{23} \) should equal the longest distance \( d_{13} \). Let's check: \[ \sqrt{34} + \sqrt{306} = \sqrt{136} = 17.32 \] Calculating the left side: \[ \sqrt{34} \approx 5.83 \] \[ \sqrt{306} \approx 17.49 \] Adding them gives us approximately \( 23.32 \), indicating that \( P_{1}, P_{2}, P_{3} \) are not equidistant along a line, thus they are not collinear.

Key Concepts

Distance Calculation in 3D Space

Calculating the distance between two points in 3D space is an extension of the Pythagorean theorem. It helps us understand how far apart two locations are in a three-dimensional framework. Here's how it works:

• For any two points \((x_1, y_1, z_1)\) and \((x_2, y_2, z_2)\), the distance \(d\) is found using the formula: \[ d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2 + (z_2-z_1)^2} \]
• This formula accounts for the differences in the x, y, and z directions, combining them to give a single straightforward measure of how far apart the points are in space.
• When calculating, remember to keep your arithmetic consistent and orderly, especially with signs—each pair of coordinates computes into a squared term, which is always positive.

This understanding of distance forms the foundation for exploring deeper spatial relationships, like collinearity.

Collinearity in 3D Mathematics

Collinearity is a property where a set of points lies on a single straight line. In 3D space, understanding collinearity involves verifying that the geometry and distances between points adhere to this linear pattern.

• When we say points are collinear, it means visually and mathematically, these points connect, forming a line. We use distance calculations to check this visual intuition.
• The key method to prove collinearity is to check if the sum of distances between two pairs equals the distance over the entire set of points. If \(P_1\), \(P_2\), and \(P_3\) are points, check if \(d_{12} + d_{23} = d_{13}\).
• If the sum doesn’t match, like \(23.32\) not equalling \(17.32\) in our sample, the points do not sit in the exact alignment necessary to form a line.

This approach confirms whether the seemingly straight path is indeed a direct linear alignment or simply an approximation.

Conducting a Mathematical Proof

Creating a mathematical proof ensures that our conclusions about concepts like distance and collinearity are clearly documented and logically sound. Here's how we layered our proof for this exercise:

• Start by applying the distance formula to calculate the separation between each point pair, accurately representing the given coordinates.
• Once the distances \(d_{12}\), \(d_{23}\), and \(d_{13}\) are computed, arrange them to verify our collinearity condition: check if the two shorter distances sum up to exactly the longer one.
• By establishing that \(d_{12} + d_{23} eq d_{13}\), we've proven with calculations that the points are not collinear. This disassembly of numeric evidence into recognizable proof showcases how precise math clarifies 3D relationships.

Proofs like these confirm or deny our hypotheses and anchor mathematical theorems in tangible, verifying numbers.