Math Problem Statement

Do the calculations

Bestimmen Sie c so, dass der Winkel zwischen der x1x2-Ebene und der Geraden g: x = r · (3, 4, c) die Größe 45° hat. Betrachten Sie alle Ursprungsgeraden, die mit der x1x2-Ebene einen Winkel von 45° bilden. Beschreiben Sie die Lage der Schnittpunkte dieser Geraden mit der Ebene E: x3 = 5.

Solution

Let's break down the problem first to ensure we handle the calculations properly.

Part (a):

We need to determine the constant cc so that the angle between the line gg and the x1x2x_1x_2-plane (i.e., the xy-plane) is 45°. The equation of the line gg is given as: x=r(34c)\vec{x} = r \cdot \begin{pmatrix} 3 \\ 4 \\ c \end{pmatrix}

To find cc, we use the fact that the angle between the direction vector of the line (34c)\begin{pmatrix} 3 \\ 4 \\ c \end{pmatrix} and the x1x2x_1x_2-plane (or xy-plane) can be computed using the formula for the angle between two vectors. In this case, the xy-plane has a normal vector along the x3x_3-axis, i.e., n=(001)\vec{n} = \begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix}.

The formula for the cosine of the angle θ\theta between the line and the plane is: cosθ=dzd\cos \theta = \frac{|\vec{d}_z|}{|\vec{d}|} where dz|\vec{d}_z| is the absolute value of the z-component of the direction vector cc, and d|\vec{d}| is the magnitude of the direction vector (34c)\begin{pmatrix} 3 \\ 4 \\ c \end{pmatrix}.

For θ=45\theta = 45^\circ, we know: cos45=22\cos 45^\circ = \frac{\sqrt{2}}{2}

Using the direction vector: d=32+42+c2=9+16+c2=25+c2|\vec{d}| = \sqrt{3^2 + 4^2 + c^2} = \sqrt{9 + 16 + c^2} = \sqrt{25 + c^2} and dz=c|\vec{d}_z| = |c|

Thus, we have the equation: c25+c2=22\frac{|c|}{\sqrt{25 + c^2}} = \frac{\sqrt{2}}{2}

Now, let's solve this equation for cc.The error occurred because we cannot directly solve equations involving the absolute value of the variable cc in this form. To proceed, we can break the equation into two cases:

  1. c0c \geq 0, so c=c|c| = c
  2. c<0c < 0, so c=c|c| = -c

Let's solve these two cases separately.The two possible solutions for cc are:

  1. c=5c = 5
  2. c=5c = -5

Thus, cc can either be 5 or -5 for the angle between the line and the x1x2x_1x_2-plane to be 45°.

Part (b):

We are asked to consider all lines through the origin that form an angle of 45° with the x1x2x_1x_2-plane. Then, we need to describe the points of intersection of these lines with the plane E:x3=5E: x_3 = 5.

These lines must have direction vectors that form a 45° angle with the xy-plane, meaning their direction vector has the form (abc)\begin{pmatrix} a \\ b \\ c \end{pmatrix}, where c=±a2+b2c = \pm\sqrt{a^2 + b^2}. Since all lines pass through the origin, their parametric form is: x=r(abc)\vec{x} = r \cdot \begin{pmatrix} a \\ b \\ c \end{pmatrix}

Now, to find the intersection with the plane x3=5x_3 = 5, set the x3x_3-coordinate of the parametric equation equal to 5: rc=5r \cdot c = 5

Solving for rr: r=5cr = \frac{5}{c}

Substituting this into the equations for x1x_1 and x2x_2: x1=5ac,x2=5bcx_1 = \frac{5a}{c}, \quad x_2 = \frac{5b}{c}

Therefore, the points of intersection are given by: (5ac,5bc,5)\left( \frac{5a}{c}, \frac{5b}{c}, 5 \right)

Would you like further details on these calculations, or clarification on any part? Here are 5 related questions:

  1. How is the angle between a line and a plane generally calculated?
  2. What would the solution be if the angle between the line and the plane was different (e.g., 30°)?
  3. How do parametric equations describe lines in 3D space?
  4. What role does the z-component cc play in the orientation of the line relative to the plane?
  5. Can the intersection points vary if the angle between the line and the plane is different?

Tip: When dealing with angles between lines and planes, always use the direction vector and normal vector to simplify calculations.

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Math Problem Analysis

Mathematical Concepts

Geometry
Analytic Geometry
Angles between Lines and Planes
Parametric Equations

Formulas

Cosine of the angle between a line and a plane: cos(θ) = |dz| / |d|
Magnitude of the direction vector: |d| = sqrt(a² + b² + c²)
Intersection of parametric line with plane: x1 = (5a)/c, x2 = (5b)/c

Theorems

Vector Geometry
Angle between Line and Plane

Suitable Grade Level

Grades 11-12 or College Level