Problem 16 - Advanced Engineering Mathematics

Question

In Problems, find symmetric equations for the line through the given points. $$ (-5,-2,-4),(1,1,2) $$

Step-by-Step Solution

Answer

Symmetric equation: \(\frac{x+5}{6} = \frac{y+2}{3} = \frac{z+4}{6}.\)

Step 1: Identify Points

We are given two points in space: Point 1: \((-5, -2, -4)\) Point 2: \((1, 1, 2)\). We need to find the symmetric equation of the line that passes through these two points.

Step 2: Find the Direction Vector

To find the direction vector, subtract the coordinates of the first point from the corresponding coordinates of the second point: \( \vec{d} = (1 - (-5), 1 - (-2), 2 - (-4)) = (6, 3, 6) \).

Step 3: Write Parametric Equations

Using one point, say \((-5, -2, -4)\), and the direction vector \((6, 3, 6)\), write the parametric equations of the line:\[ x = -5 + 6t, \]\[ y = -2 + 3t, \]\[ z = -4 + 6t, \]where \(t\) is a parameter.

Step 4: Write Symmetric Equations

Transform the parametric equations into symmetric form:Equate each parametric equation to the common parameter \(t\):\[ t = \frac{x + 5}{6} = \frac{y + 2}{3} = \frac{z + 4}{6}. \]These are the symmetric equations of the line through the points.

Key Concepts

Direction Vector

Understanding the concept of a direction vector is fundamental when dealing with lines in space. A direction vector provides a way to describe the direction and orientation of a line. It is essentially a vector that points from one point to another along the line. In our example, the direction vector was calculated using two given points,
• Point 1: (-5, -2, -4)
• Point 2: (1, 1, 2)
To find the direction vector, subtract the coordinates of the first point from the corresponding coordinates of the second point:\[\vec{d} = (1 - (-5), 1 - (-2), 2 - (-4)) = (6, 3, 6).\]This resulting vector (6, 3, 6) provides the orientation of the line. The direction vector is crucial not only in defining a line but also in constructing both parametric and symmetric equations. Without it, we wouldn’t know how the line stretches through space.

Parametric Equations

Parametric equations are a set of equations that express the coordinates of the points on a geometric object as functions of a single parameter. This is particularly useful in three-dimensional space, as it allows us to track all points along a line efficiently.
In our case, the parameter trepresents how far along the line we are from a starting point. Using the point (-5, -2, -4) and the direction vector (6, 3, 6),we write the parametric equations for the line:

• \(x = -5 + 6t\)
• \(y = -2 + 3t\)
• \(z = -4 + 6t\)

These equations mean that as tvaries, we move along the line. Each equation describes how a coordinate changes in relation to the parameter. Parametric equations are extremely helpful in computer graphics and simulations, where moving along a path or line in space requires tracking transitions smoothly.

Points in Space

Points in space are fundamental elements in geometry, especially when it comes to understanding lines and other geometric shapes. In three-dimensional space, each point is defined by three coordinates, \(x, y, \)and \(z\),that tell us the position relative to an origin. In the original exercise provided, we started with two points:

• Point 1: (-5, -2, -4)
• Point 2: (1, 1, 2)

These points set the boundary conditions for forming the line. Using points in space, you can determine not just lines but also planes and volumes by understanding how they relate and interact. By finding the direction vector and forming the parametric equations from these points, the line was defined that's threading through both points. Thus, in geometry, identifying key points is always the starting point of solving shape-based problems.