Problem 18 - Advanced Engineering Mathematics

Question

In Problems, find symmetric equations for the line through the given points. $$ \left(\frac{5}{6},-\frac{1}{4}, \frac{1}{5}\right),\left(\frac{1}{3}, \frac{3}{8},-\frac{1}{10}\right) $$

Step-by-Step Solution

Answer

The symmetric equations are \(\frac{x - \frac{5}{6}}{-\frac{1}{2}} = \frac{y + \frac{1}{4}}{\frac{5}{8}} = \frac{z - \frac{1}{5}}{-\frac{3}{10}} \).

Step 1: Identify Points

We are given two points: \(P_1\left(\frac{5}{6}, -\frac{1}{4}, \frac{1}{5}\right)\) and \(P_2\left(\frac{1}{3}, \frac{3}{8}, -\frac{1}{10}\right)\). These will be used to find the direction of the line.

Step 2: Calculate Direction Vector

The direction vector \( \mathbf{d} \) of the line can be found by calculating \(P_2 - P_1\):\[\mathbf{d} = \left( \frac{1}{3} - \frac{5}{6}, \frac{3}{8} - (-\frac{1}{4}), -\frac{1}{10} - \frac{1}{5} \right)\]Simplifying each component, we get:\[\mathbf{d} = \left(-\frac{1}{2}, \frac{5}{8}, -\frac{3}{10} \right)\].

Step 3: Write Parametric Equations

Using the direction vector and a point on the line, we can write the parametric equations for the line:\[ x = \frac{5}{6} + t\left(-\frac{1}{2}\right) \]\[ y = -\frac{1}{4} + t\left(\frac{5}{8}\right) \]\[ z = \frac{1}{5} + t\left(-\frac{3}{10}\right) \]where \(t\) is a parameter.

Step 4: Convert to Symmetric Equations

The symmetric equations can be obtained by eliminating the parameter \(t\) from the parametric equations:\[ \frac{x - \frac{5}{6}}{-\frac{1}{2}} = \frac{y + \frac{1}{4}}{\frac{5}{8}} = \frac{z - \frac{1}{5}}{-\frac{3}{10}} \].These represent the symmetric form of the line through the points.

Key Concepts

Direction Vector

When determining the path of a line in 3D coordinate geometry, the direction vector plays a crucial role. It tells us the direction the line is heading. To find a direction vector for a line passing through two points, you subtract the coordinates of the initial point from the corresponding coordinates of the terminal point.

For the given example, the direction vector \( \mathbf{d} \) was calculated by subtracting point \( P_1 \) from point \( P_2 \), resulting in \( \mathbf{d} = \left(-\frac{1}{2}, \frac{5}{8}, -\frac{3}{10} \right) \). This vector describes the direction and the orientation of our line.

Remember, a direction vector can be scaled by multiplying it with a scalar, and the line's orientation will remain the same. This means that any positive or negative multiple of the direction vector still points in the same or directly opposite direction.

Parametric Equations

The parametric equations of a line in three-dimensional space are derived using a point on the line and its direction vector. These equations express the line in terms of a parameter \( t \), which accounts for every point along the line.

For example, using our earlier direction vector \( \mathbf{d} = \left(-\frac{1}{2}, \frac{5}{8}, -\frac{3}{10} \right) \) and the point \( P_1\left(\frac{5}{6}, -\frac{1}{4}, \frac{1}{5}\right) \), we constructed the equations:

• \( x = \frac{5}{6} + t\left(-\frac{1}{2}\right) \)
• \( y = -\frac{1}{4} + t\left(\frac{5}{8}\right) \)
• \( z = \frac{1}{5} + t\left(-\frac{3}{10}\right) \)

These parametric equations enable you to find any specific point on the line by choosing a value for \( t \), allowing for detailed explorations of 3D geometry lines.

3D Coordinate Geometry

3D coordinate geometry extends traditional 2D analysis into the third dimension, adding both complexity and real-world applicability. This field explores lines, planes, and surfaces in a three-dimensional space.

In our exercise, we dealt with finding symmetric equations, which essentially eliminate the parameter from the parametric form, giving us a direct characterization of the line in space. These equations highlight the relationships between the x, y, and z coordinates without explicitly involving time or sequence (like parameter \( t \)).

• Symmetric equations in the example: \( \frac{x - \frac{5}{6}}{-\frac{1}{2}} = \frac{y + \frac{1}{4}}{\frac{5}{8}} = \frac{z - \frac{1}{5}}{-\frac{3}{10}} \).

These types of equations provide another perspective to analyze the line, offering unique computational benefits, especially when examining intersections or distances in 3D space.