Problem 62 - Thomas Calculus

Question

Show that the point \(P(3,1,2)\) is equidistant from the points \(A(2,-1,3)\) and \(B(4,3,1)\) .

Step-by-Step Solution

Answer

Point P is equidistant from points A and B because both distances PA and PB are equal to \(\sqrt{6}\).

Step 1: Determine the distance formula in 3D

The distance between two points \((x_1, y_1, z_1)\) and \((x_2, y_2, z_2)\) in three-dimensional space is given by the formula \(\sqrt{(x_2-x_1)^2 + (y_2-y_1)^2 + (z_2-z_1)^2}\). We will use this formula to find the distances from point \(P(3,1,2)\) to points \(A(2,-1,3)\) and \(B(4,3,1)\).

Step 2: Calculate the distance between P and A

Using the distance formula, calculate \(PA\): \[PA = \sqrt{(3-2)^2 + (1+1)^2 + (2-3)^2}\]Simplifying, we get:\[PA = \sqrt{1^2 + 2^2 + (-1)^2} = \sqrt{1 + 4 + 1} = \sqrt{6}\]

Step 3: Calculate the distance between P and B

Using the distance formula, calculate \(PB\): \[PB = \sqrt{(3-4)^2 + (1-3)^2 + (2-1)^2}\]Simplifying, we get:\[PB = \sqrt{(-1)^2 + (-2)^2 + 1^2} = \sqrt{1 + 4 + 1} = \sqrt{6}\]

Step 4: Compare the distances

Since both distances \(PA\) and \(PB\) are equal to \(\sqrt{6}\), point \(P(3,1,2)\) is equidistant from both points \(A(2,-1,3)\) and \(B(4,3,1)\).

Key Concepts

Equidistant Points

Equidistant points in geometry are points that have the same distance from two or more other points. In our exercise, we need to demonstrate that point \( P(3,1,2) \) is equidistant from two other points, \( A(2,-1,3) \) and \( B(4,3,1) \). This means the length of the line segments \( PA \) and \( PB \) must be identical.To determine if a point is equidistant, you'll calculate the respective distances and verify that they are equal. If they are, then the point in question is equidistant from the others. In this exercise, we showed that both distances from point \( P \) to points \( A \) and \( B \) are \( \sqrt{6} \). This confirms point \( P \) is equidistant from \( A \) and \( B \).Understanding equidistant points is fundamental in coordinate geometry, as it allows us to establish relationships and solve problems in multi-dimensional spaces.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, involves studying geometry using a coordinate system. In 3D space, we describe points with three coordinates: \( (x, y, z) \). This expands upon the familiar 2D plane by adding a third dimension, allowing for a greater range of geometric problems.

• Points: Defined by an ordered triplet of numbers.
• Distances: Calculated using the 3D distance formula.
• Lines and Surfaces: Described by equations derived from coordinates.

With coordinate geometry, we are able to solve more complex problems by transforming geometric problems into algebraic expressions and equations. By plotting and calculating points like \( P, A, \) and \( B \) in three-dimensional space, we can also find relationships like equidistance more precisely.Coordinate geometry bridges algebra and geometry, equipping us with tools to solve a variety of mathematical problems. It is crucial in applications such as physics, engineering, and computer graphics.

3D Distance Calculation

The 3D distance calculation involves finding the straight-line distance between two points in three-dimensional space. The formula is an extension of the Pythagorean theorem into three dimensions, given by:\[D = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2 + (z_2-z_1)^2}\]where \((x_1, y_1, z_1)\) and \((x_2, y_2, z_2)\) are the coordinates of the two points.Let's break this down:

• Subtract the coordinates of the starting point from the coordinates of the endpoint for each axis.
• Square each of these differences.
• Add these squared differences together.
• Take the square root of the sum to find the distance.

Using this formula, we calculated both \( PA \) and \( PB \) as \( \sqrt{6} \). Mastering this calculation is critical for working with 3D models, navigation systems, and understanding spatial relationships in various fields of study. It’s an invaluable tool in not only math, but also in real-world applications where precise measurements are crucial.