Problem 15 - Advanced Engineering Mathematics

Question

In Problems, find symmetric equations for the line through the given points. $$ (4,2,1),(-7,2,5) $$

Step-by-Step Solution

Answer

The symmetric equations are \( \frac{x - 4}{-11} = \frac{z - 1}{4} \) and \( y = 2 \).

Step 1: Find the Direction Vector

To find the symmetric equations of the line passing through two points, we first need to determine the direction vector of the line. Given two points \( (4, 2, 1) \) and \( (-7, 2, 5) \), the direction vector \( \mathbf{d} \) can be found by subtracting the coordinates of the first point from the second point:\[ \mathbf{d} = (-7 - 4, 2 - 2, 5 - 1) = (-11, 0, 4) \]

Step 2: Write the Parametric Equations

Next, use the direction vector to write the parametric equations of the line. The point \( (4, 2, 1) \) can be used as an initial point, and \( t \) is a parameter that varies over all real numbers.The parametric equations are:\[ x = 4 - 11t \]\[ y = 2 \]\[ z = 1 + 4t \]

Step 3: Derive Symmetric Equations

To derive symmetric equations from the parametric equations, solve each for the parameter \( t \):From \( x = 4 - 11t \), solve for \( t \):\[ t = \frac{4 - x}{11} \]From \( z = 1 + 4t \), solve for \( t \):\[ t = \frac{z - 1}{4} \]Since \( y = 2 \) remains constant, the symmetric equations of the line are:\[ \frac{4 - x}{11} = \frac{z - 1}{4} \]\[ y = 2 \]

Step 4: Assemble the Final Symmetric Equation

Now combine the previous results into the final form of the symmetric equations:The symmetric equations for the line in terms of \( x \) and \( z \) are:\[ \frac{x - 4}{-11} = \frac{z - 1}{4} \]And note \( y = 2 \) as part of the line's characteristics.

Key Concepts

Direction Vector

When discussing lines, especially in 3D geometry, the concept of a direction vector is crucial. A direction vector helps define the line's orientation in space. It indicates the line's direction as it extends infinitely.
For a line passing through two points, such as

• Point A: \( (4, 2, 1) \)
• Point B: \( (-7, 2, 5) \)

we calculate the direction vector by performing vector subtraction. This involves subtracting the coordinates of Point A from Point B:
\[ \mathbf{d} = (-7 - 4, 2 - 2, 5 - 1) = (-11, 0, 4) \]
The result is \( \mathbf{d} = (-11, 0, 4) \). This vector indicates how far and in which direction you move from Point A to Point B. With this, we now have crucial information for expressing the line's equation.

Parametric Equations

Parametric equations express a line in terms of a parameter, usually denoted as \( t \). These equations allow us to represent lines through a chosen reference point and a direction vector.

• Initial point: \( (4, 2, 1) \)
• Direction vector: \( (-11, 0, 4) \)

Each coordinate of the line can now be expressed as a function of \( t \). The parametric equations for our line are:
\[ x = 4 - 11t \] \[ y = 2 \] \[ z = 1 + 4t \]
Here, \( t \) changes, allowing these equations to describe every point along the line in 3D space. Knowing parametric equations is useful, especially for describing trajectory and position at different times.

3D Geometry

3D geometry extends the familiar 2D concepts into three dimensions, adding depth to height and width. It deals with objects like lines, planes, and shapes that exist in this three-dimensional space.
A line in 3D is not just a straight path but can be described comprehensively using its direction vector and parametric equations. The addition of the z-coordinate in 3D means lines can tilt and orient in ways not possible in 2D.
This creates a more sophisticated framework for understanding spatial relationships and can be applied in various fields such as physics, engineering, and computer graphics.

Vector Subtraction

Vector subtraction is the process of finding the difference between two vectors. It's an essential tool in geometry and physics. In our exercise, we used it to determine the line's direction.
To perform vector subtraction, you subtract each corresponding component of one vector from another. This gives you a new vector that shows how much you move in each axis from one vector's tip to the other's tip.
For the points \( (4, 2, 1) \) and \( (-7, 2, 5) \), subtracting gives us the direction vector \( (-11, 0, 4) \).
This new vector points from the first point to the second, encapsulating both magnitude and direction. Vector subtraction thus becomes a cornerstone technique when assessing relationships between points in a space.