Math Problem Statement

Solve the given tasks related to plane equations and lines in 3D space.

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.

  1. Part a:
    The plane e1e_1 is parallel to the x1x3x_1 x_3-plane and passes through the point P(040)P(0|4|0).

    • A plane parallel to the x1x3x_1 x_3-plane has a normal vector parallel to the x2x_2-axis, i.e., the normal vector is n=(0,1,0)\mathbf{n} = (0, 1, 0).
    • Using the point-normal form of a plane equation: n1(xx0)+n2(yy0)+n3(zz0)=0n_1(x - x_0) + n_2(y - y_0) + n_3(z - z_0) = 0, we get the equation for this plane as:
      1(x24)=0or simplyx2=4.1 \cdot (x_2 - 4) = 0 \quad \text{or simply} \quad x_2 = 4.
  2. Part b:
    The plane e2e_2 is perpendicular to the x1x_1-axis and contains the point P(123)P(1|2|3).

    • A plane perpendicular to the x1x_1-axis has a normal vector n=(1,0,0)\mathbf{n} = (1, 0, 0).
    • Using the same form of equation as above and the given point, the equation for this plane is:
      1(x11)=0or simplyx1=1.1 \cdot (x_1 - 1) = 0 \quad \text{or simply} \quad x_1 = 1.
  3. Part c:
    The plane e3e_3 passes through the origin and is perpendicular to the line passing through the points P(321)P(3|2|1) and Q(201)Q(2|0|1).

    • The direction vector of the line through PP and QQ is d=PQ=(1,2,0)\vec{d} = P - Q = (1, 2, 0).
    • 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.

  1. Part a:
    To define a plane uniquely by a point PP and a line g:x=a+λug: \vec{x} = \vec{a} + \lambda \cdot \vec{u}, the point PP must not lie on the line gg. Also, the vector connecting the point PP and any point on the line must not be parallel to the direction vector of the line u\vec{u}. The condition for this is that the vector Pa\vec{P} - \vec{a} must not be collinear with u\vec{u}.

  2. Part b:
    The equation of the plane in parametric form can be written as:
    x=a+λu+μ(Pa).\vec{x} = \vec{a} + \lambda \vec{u} + \mu (\vec{P} - \vec{a}).

Aufgabe 3 (Problem 3):

The task is to match the given diagrams of planes to their corresponding equations.

  • Diagram 1: The plane is vertical along x2x_2 and passes through a certain area in the x1x3x_1 x_3-plane. A likely equation is:
    4x13x3=124x_1 - 3x_3 = 12 (option a).

  • Diagram 2: The plane is vertical along the x3x_3-axis and has an intercept on the x2x_2-axis. A likely equation is:
    x24x3=8x_2 - 4x_3 = 8 (option b).

  • Diagram 3: The plane is parallel to the x2x3x_2 x_3-plane and cuts the x1x_1-axis at a specific point. A possible equation is:
    x3=2x_3 = 2 (option c).

  • Diagram 4: The plane is parallel to the x1x2x_1 x_2-plane and lies at a fixed height on the x3x_3-axis. A possible equation is:
    x3=0x_3 = 0 (option f).


Let me know if you need detailed clarifications on any of these points!

Here are 5 related questions for further practice:

  1. How do you derive the normal vector of a plane from a given line and point?
  2. Can a plane be uniquely determined by two parallel lines?
  3. How do you find the intersection line of two planes?
  4. How can you generalize the equation of a plane passing through three points?
  5. 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)