(1 Point) The Planes 5x + 3y + 5z = -19 And 2z - 5y = 17 Are Not Parallel, So They Must Intersect Along
Understanding the intersection of planes is a fundamental concept in three-dimensional geometry. When dealing with planes in space, determining whether they are parallel, intersecting, or coincident provides insight into their spatial relationships. In this article, we will analyze the specific planes given by the equations:
- Plane 1: 5x + 3y + 5z = -19
- Plane 2: 2z - 5y = 17
We will explore why these planes are not parallel and what their intersection looks like. The goal is to understand how to find the line of intersection between two non-parallel planes, interpret their geometric relationship, and apply this knowledge to related problems.
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Understanding Planes in Three-Dimensional Space
The General Equation of a Plane
In three-dimensional Cartesian coordinates, a plane can be represented by a linear equation of the form:
\[ ax + by + cz + d = 0 \]
where
- \(a\), \(b\), and \(c\) are the coefficients representing the normal vector to the plane,
- \(d\) is the constant term.
The normal vector, \(\vec{n} = (a, b, c)\), is perpendicular to the surface of the plane.
Conditions for Parallelism, Coincidence, and Intersection
- Parallel Planes: Two planes are parallel if their normal vectors are scalar multiples of each other, and they are not coincident.
- Coincident Planes: Two planes are coincident if their equations are scalar multiples of each other.
- Intersecting Planes: Two planes intersect along a line if their normal vectors are not scalar multiples; in this case, they are neither parallel nor coincident.
- Plane 1: \(5x + 3y + 5z = -19\)
- Plane 2: \(2z - 5y = 17\)
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Analyzing the Planes: Are They Parallel?
Normal Vectors of the Given Planes
- For Plane 1: \(5x + 3y + 5z = -19\)
\[
\vec{n}_1 = (5, 3, 5)
\]
- For Plane 2: \(2z - 5y = 17\)
Rewrite as:
\[
0x - 5y + 2z = 17
\]
The normal vector is:
\[
\vec{n}_2 = (0, -5, 2)
\]
Checking for Parallelism
Two vectors are scalar multiples if there exists a scalar \(k\) such that:
\[
\vec{n}1 = k \cdot \vec{n}2
\]
Check each component:
- \(5 = k \times 0 \Rightarrow 5 = 0 \Rightarrow\) Contradiction
Since the first component of \(\vec{n}2\) is zero, and the first component of \(\vec{n}1\) is 5, they cannot be scalar multiples. Therefore, the planes are not parallel.
This confirms that the planes must intersect along a line, which is the common intersection line of the two planes.
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Finding the Line of Intersection
To find the line along which the planes intersect, we need to:
- Solve the system of equations representing the planes.
- Express the solution as a parametric equation of a line.
Step 1: Write the system of equations
\[
\begin{cases}
5x + 3y + 5z = -19 \quad \text{(Equation 1)} \\
2z - 5y = 17 \quad \text{(Equation 2)}
\end{cases}
\]
---
Step 2: Express variables in terms of a parameter
Choose a variable to parameterize. Let’s choose \(y\) as the free parameter, say:
\[
y = t
\]
Now, solve Equation 2 for \(z\):
\[
2z - 5t = 17
\]
\[
2z = 17 + 5t
\]
\[
z = \frac{17 + 5t}{2}
\]
Next, substitute \(y = t\) and \(z\) into Equation 1:
\[
5x + 3t + 5 \left( \frac{17 + 5t}{2} \right) = -19
\]
Multiply out:
\[
5x + 3t + \frac{5 \times (17 + 5t)}{2} = -19
\]
\[
5x + 3t + \frac{85 + 25t}{2} = -19
\]
Multiply both sides by 2 to clear the denominator:
\[
2 \times 5x + 2 \times 3t + 85 + 25t = -38
\]
\[
10x + 6t + 85 + 25t = -38
\]
Combine like terms:
\[
10x + (6t + 25t) + 85 = -38
\]
\[
10x + 31t + 85 = -38
\]
Solve for \(x\):
\[
10x = -38 - 85 - 31t
\]
\[
10x = -123 - 31t
\]
\[
x = \frac{-123 - 31t}{10}
\]
---
Step 3: Write the parametric equations of the line
Using the parameter \(t\), the parametric equations are:
\[
\boxed{
\begin{cases}
x = \frac{-123 - 31t}{10} \\
y = t \\
z = \frac{17 + 5t}{2}
\end{cases}
}
\]
or, more compactly, the line of intersection can be expressed as:
\[
\boxed{
\begin{aligned}
x &= -12.3 - 3.1 t \\
y &= t \\
z &= 8.5 + 2.5 t
\end{aligned}
}
\]
where \(t\) is any real number.
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Geometric Interpretation of the Line of Intersection
The line of intersection between the two planes is an infinite line passing through space, defined by the parametric equations derived above. This line contains all points \((x, y, z)\) that satisfy both plane equations simultaneously.
This intersection line is significant because:
- It represents the set of all points lying on both planes.
- It helps in understanding the spatial relationship between the planes.
- It can serve as a foundation for solving more complex geometric problems involving planes in three dimensions.
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Applications of Plane Intersections in Real Life
Understanding the intersection of planes has numerous practical applications in various fields, including:
- Engineering and Architecture: Designing structures requires understanding how different surfaces intersect.
- Computer Graphics: Rendering 3D scenes involves calculating intersections of objects.
- Geology: Modeling fault lines and strata involves analyzing intersecting planes.
- Robotics: Navigating in 3D space often involves understanding plane intersections for obstacle avoidance.
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Summary and Key Takeaways
- Two planes in space are not parallel if their normal vectors are not scalar multiples.
- When planes are not parallel, they intersect along a line.
- To find the line of intersection:
- Solve the system of plane equations.
- Express variables in terms of a free parameter.
- Write the parametric equations of the line.
- The normal vectors of the planes in our example confirm they are not parallel, and the derived parametric equations define their intersection line.
Conclusion
The analysis of the planes given by \(5x + 3y + 5z = -19\) and \(2z - 5y = 17\) demonstrates a fundamental concept in three-dimensional geometry: intersecting planes. By examining their normal vectors, confirming they are not scalar multiples, and solving their system of equations, we find that they intersect along a specific line described parametrically.
Understanding such intersections is crucial in various scientific and engineering disciplines. It allows professionals to visualize complex spatial relationships, solve real-world problems, and design systems with precision.
Whether you are a student learning about geometry or a professional applying these concepts in practical scenarios, mastering the process of analyzing plane intersections is essential for a comprehensive understanding of three-dimensional space.
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Meta Description: Discover how to determine and find the line of intersection between two non-parallel planes given by \(5x + 3y + 5z = -19\) and \(2z - 5y = 17\). Learn the step-by-step process, parametric equations, and geometric interpretation in this detailed guide.
