Problem 22 - Advanced Engineering Mathematics

Question

Find the distance between the given points. $$ (-1,-3,5),(0,4,3) $$

Step-by-Step Solution

Answer

The distance between the points is \(3\sqrt{6}\).

Step 1: Identify the Coordinates

First, identify the coordinates of the two points given. The first point is \((-1, -3, 5)\) and the second point is \((0, 4, 3)\).

Step 2: Use the Distance Formula

The distance \(d\) between two points \((x_1, y_1, z_1)\) and \((x_2, y_2, z_2)\) in three-dimensional space can be calculated using the formula:\[d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}\]

Step 3: Substitute the Coordinates

Substitute the given coordinates into the formula:\[d = \sqrt{(0 - (-1))^2 + (4 - (-3))^2 + (3 - 5)^2}\]

Step 4: Calculate Each Difference

Calculate the differences for each coordinate:- \(0 - (-1) = 1\)- \(4 - (-3) = 7\)- \(3 - 5 = -2\)

Step 5: Square Each Difference

Now, square each of the differences obtained:- \(1^2 = 1\)- \(7^2 = 49\)- \((-2)^2 = 4\)

Step 6: Sum the Squares

Sum the results of the squared values:\[1 + 49 + 4 = 54\]

Step 7: Calculate the Square Root

Finally, take the square root of the sum from the previous step to find the distance:\[\sqrt{54} = 3\sqrt{6}\]

Key Concepts

Coordinate Geometry

Coordinate geometry is a branch of mathematics that introduces geometric figures like points, lines, and circles using a coordinate system. This allows us to apply algebraic methods to geometry.

• By using coordinates, we can easily perform calculations like finding distances between points, determining midpoints, and analyzing slopes.
• Each point in space is defined by a set of coordinates based on its position on horizontal and vertical lines, and in three-dimensional space, this extends to a third coordinate representing depth.

Coordinate geometry not only simplifies complex geometric problems but also enables the solving of problems via algebraic expressions. In this context, the coordinates of points such as (-1,-3,5) and (0,4,3) can represent locations in three-dimensional space, which allows us to use specific formulas to derive the spatial relationship between them. Understanding how to work with coordinate systems is fundamental for transitioning into more complex geometry and calculus problems.

3-Dimensional Distance

Finding distances between points in three-dimensional space is a valuable skill in coordinate geometry. When working in three dimensions, each point is defined by three coordinates:

• The x-coordinate, representing horizontal placement.
• The y-coordinate, indicating vertical placement.
• The z-coordinate, defining depth.

To calculate the distance between the points like (-1,-3,5) and (0,4,3), we extend the familiar distance formula from two dimensions to include the z-coordinate:\[d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}\]Here, we use this formula to compute the straight-line distance. After finding the differences in each coordinate, squaring them, and then taking the sum, the step-by-step solution completes by taking the square root. This method gives the precise distance between any two points in 3D space, revealing how mathematics can be applied to measure real-world distances accurately.

Mathematical Problem Solving

Mathematical problem solving in the context of coordinate geometry encourages logical thinking and systematic approaches. It involves breaking down complex tasks into smaller, manageable steps.

• Firstly, identify what is being asked and gather all necessary information.
• Use known formulas and understand when and how to apply them.
• Solve systematically, checking each step's result as you progress.

In the example provided, solving for the distance between two points involves several clear steps: 1. Identify coordinates for each point. 2. Apply the 3D distance formula by substituting these coordinates. 3. Perform arithmetic calculations for differences and their squares. 4. Sum results and take the square root to find the distance. Such structured problem solving not only aids in finding solutions effectively but also enhances comprehensive analytical skills, crucial for more advanced mathematics and practical applications in sciences, engineering, and technology.