Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 4: Problem 2

Berechnen Sie die Schnittpunk te der Ebene \(2 x-5 y+3 z=1\) mit den Koordinatenachsen. Welche der folgenden Punkte licgen in der Ebene, welche unterhalb brw. oberhalb der Fbene? $$ A=(1 ; 2 ; 3), \quad B=(2 ;-3 ;-4), \quad C=(-4 ; 2 ; 0), \quad D=(0 ; 0 ; 0) $$

Short Answer

Expert verified
Intersections: (0.5, 0, 0), (0, -0.2, 0), (0, 0, 0.333). Points: A on, B above, C below, D below.

Step by step solution

01

- Calculate the intersection with the x-axis

To find the intersection with the x-axis, set y and z to 0 and solve for x in the plane equation. \[ 2x - 5(0) + 3(0) = 1 \]\[ 2x = 1 \]\[ x = \frac{1}{2} \]So, the intersection with the x-axis is \( \left( \frac{1}{2}, 0, 0 \right) \).
02

- Calculate the intersection with the y-axis

To find the intersection with the y-axis, set x and z to 0 and solve for y in the plane equation. \[ 2(0) - 5y + 3(0) = 1 \]\[ -5y = 1 \]\[ y = -\frac{1}{5} \]So, the intersection with the y-axis is \( \left( 0, -\frac{1}{5}, 0 \right) \).
03

- Calculate the intersection with the z-axis

To find the intersection with the z-axis, set x and y to 0 and solve for z in the plane equation. \[ 2(0) - 5(0) + 3z = 1 \]\[ 3z = 1 \]\[ z = \frac{1}{3} \]So, the intersection with the z-axis is \( \left( 0, 0, \frac{1}{3} \right) \).
04

- Check point A

For point A (1, 2, 3), substitute into the plane equation: \[ 2(1) - 5(2) + 3(3) = 2 - 10 + 9 = 1 \]Point A lies on the plane.
05

- Check point B

For point B (2, -3, -4), substitute into the plane equation: \[ 2(2) - 5(-3) + 3(-4) = 4 + 15 - 12 = 7 \]Point B lies above the plane.
06

- Check point C

For point C (-4, 2, 0), substitute into the plane equation: \[ 2(-4) - 5(2) + 3(0) = -8 - 10 = -18 \]Point C lies below the plane.
07

- Check point D

For point D (0, 0, 0), substitute into the plane equation: \[ 2(0) - 5(0) + 3(0) = 0 \]Point D lies below the plane.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Coordinate Axes Intersections
The intersections of a plane with the coordinate axes are essential foundational points in geometry. To find these intersections, you set the other two coordinates to zero and solve for the remaining variable in the plane's equation. Let's break it down:

● **X-axis Intersection:** When y and z are set to zero, solving the given plane equation simplifies to finding x.
● **Y-axis Intersection:** When x and z are set to zero, solving the equation simplifies to finding y.
● **Z-axis Intersection:** When x and y are set to zero, solving the equation simplifies to finding z.

Understanding this helps in visualizing how a plane cuts through three-dimensional space.
Point Position Relative to Plane
Determining whether a point lies on, above, or below the plane involves substituting the point's coordinates into the plane equation. Here's the trick:

1. Substitute the point's coordinates into the plane's equation.
2. Solve for the value.

For example, with our plane equation \(2x - 5y + 3z = 1\) and point A(1, 2, 3):
here's the substitution: \(2(1) - 5(2) + 3(3) = 2 - 10 + 9 = 1\).
This shows point A lies on the plane.

If a point does not satisfy the plane equation precisely (i.e., it results in a number other than the plane's constant term), it's either above or below, depending on the sign of the resultant value.
Plane Equation Solutions
Understanding plane equations is key. A plane equation usually has the format \(Ax + By + Cz = D\). Each coefficient (A, B, C) influences the plane's orientation.

To solve, you:
• Set coordinates to zero for one of the axes.
• Solve the simplified equation for the remaining variable.

For instance, finding the intersection with the x-axis in our plane equation \(2x - 5y + 3z = 1\) , we set y and z to zero:
\(2x = 1\), so \(x = \frac{1}{2}\). This intersection point, therefore, is \(\frac{1}{2}, 0, 0\).
Analytical Geometry Methods
Analytical geometry allows us to use algebraic methods to solve geometric problems. This field bridges algebra and geometry via coordinate systems.

Key methods include:
• **Substitution:** Plugging coordinates into equations to test point positions.
• **Solving systems:** Making use of both the plane equation and coordinate values to find intersections.

Through these methods, we step from abstract concepts to concrete solutions, making it easier to visualize and understand the geometric relationships in three-dimensional space.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Berechnen Sie unter Verwendung des totalen Differentials die Oberfl?chen?nderung. \(\Delta O \approx d O\) eines Zylinders mit Boden und Deckel, dessen Radius \(r=10 \mathrm{~cm}\) um \(5 \%\) vergrö\betaert und dessen H?he \(h=25 \mathrm{~cm}\) gleichreitig um \(2 \%\) werkleinert wurde und vergleichen Sie diesen N?herungswert mit dem exakten Wert \(\triangle O\) exakt.

Bilden Sie die partiellen Ableitwngen 1. Ordnung der jeweiligen Funktion \(z=f(x ; y)\) mit \(x=x(u ; v), y=y(u ; v)\) nach den Parametern \(u\) bzw, \(v\) unter Verwendung der Kettenregel: a) \(z=\tan (x+y)\) mit \(x=u^{2}+v, y=u^{2}-v\) b) \(\quad z=x^{2} y^{3}+x^{3} y^{2} \quad\) mit \(x=u+v, y=u-v\)

Zeigen Sie: Durch Rotation einer Ellipse mit den Halbachsen \(a\) und \(b\) um die 2-Achse entsteht ein Rotationsellipsoid vom Volumen \(V_{z}=\frac{4}{3} \pi a^{2} b\).

Berechnen Sie die Fl?che zwischen der Parabel \(y=x^{2}\) und der Geraden \(y=-x+6\)

Differenzieren Sie die jeweilige Funktion \(z=f(x: y)\) mit \(x=x(t), y=y(t)\) nach dem Parameter \(t\) unter Verwendung der Kettenregel: a) \(z=\mathrm{e}^{x y} \quad\) mit \(x=t^{2}, y=t\) b) \(z=x \cdot \ln y \quad\) mit \(\quad x=\sin t, y=\cos t\) c) \(z=x^{2} \cdot \sin (2 y)\) mit \(x=t^{2}, y=t^{3}\)
See all solutions

Recommended explanations on Physics Textbooks

Dynamics

Read Explanation

Astrophysics

Read Explanation

Electricity

Read Explanation

Further Mechanics and Thermal Physics

Read Explanation

Electrostatics

Read Explanation

Translational Dynamics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.

Sign-up for free