Math Problem Statement
Solution
Let's go through each question in your image step by step.
Aufgabe 1 (Problem 1):
You are asked to provide a normal equation and a possible coordinate equation for different planes.
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Part a:
The plane is parallel to the -plane and passes through the point .- A plane parallel to the -plane has a normal vector parallel to the -axis, i.e., the normal vector is .
- Using the point-normal form of a plane equation: , we get the equation for this plane as:
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Part b:
The plane is perpendicular to the -axis and contains the point .- A plane perpendicular to the -axis has a normal vector .
- Using the same form of equation as above and the given point, the equation for this plane is:
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Part c:
The plane passes through the origin and is perpendicular to the line passing through the points and .- The direction vector of the line through and is .
- The normal vector of the plane is perpendicular to this direction vector, so we can use it in the cross-product to form the equation of the plane. Using the given points and the general plane equation method will give you the result.
Aufgabe 2 (Problem 2):
Here, you're asked to define a plane by a point and a line, not just by three points.
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Part a:
To define a plane uniquely by a point and a line , the point must not lie on the line . Also, the vector connecting the point and any point on the line must not be parallel to the direction vector of the line . The condition for this is that the vector must not be collinear with . -
Part b:
The equation of the plane in parametric form can be written as:
Aufgabe 3 (Problem 3):
The task is to match the given diagrams of planes to their corresponding equations.
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Diagram 1: The plane is vertical along and passes through a certain area in the -plane. A likely equation is:
(option a). -
Diagram 2: The plane is vertical along the -axis and has an intercept on the -axis. A likely equation is:
(option b). -
Diagram 3: The plane is parallel to the -plane and cuts the -axis at a specific point. A possible equation is:
(option c). -
Diagram 4: The plane is parallel to the -plane and lies at a fixed height on the -axis. A possible equation is:
(option f).
Let me know if you need detailed clarifications on any of these points!
Here are 5 related questions for further practice:
- How do you derive the normal vector of a plane from a given line and point?
- Can a plane be uniquely determined by two parallel lines?
- How do you find the intersection line of two planes?
- How can you generalize the equation of a plane passing through three points?
- What are the conditions for two planes to be perpendicular to each other?
Tip: Always remember to check if vectors are parallel or perpendicular when working with planes and lines.
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Math Problem Analysis
Mathematical Concepts
Plane equations
Normal vectors
Line and plane interactions
Parametric forms
Formulas
Point-normal form of a plane equation: n1(x - x0) + n2(y - y0) + n3(z - z0) = 0
Parametric form of a plane equation: x = a + λu + μ(v)
Vector form to find direction vector: PQ = Q - P
Theorems
Equation of a plane passing through a point with a given normal vector
Conditions for a point and a line to determine a plane
Suitable Grade Level
University-level mathematics (linear algebra, vector calculus)
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