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Chapter 12: Problem 5

Describe and sketch the surface in \(\mathbb{R}^{3}\) represented by the equation \(x+y=2\)

Short Answer

Expert verified
The surface is a plane parallel to the z-axis, intersecting the xy-plane at the line x + y = 2.

Step by step solution

01

Identify the Equation Type

The equation given is of the form of a plane in \( ext{3-D}\) space. Generally, equations in the format \(ax + by + cz = d\) represent planes.
02

Rewrite the Equation in Standard Form

The equation given is \(x + y = 2\). To express it in a more recognizable plane form, assume the equation \(x + y + 0z = 2\). This indicates that the term with \(z\) is zero.
03

Identify the Direction of the Plane

Since the coefficient of \(z\) is zero, the plane is parallel to the \(z\)-axis. This means the plane extends infinitely along the \(z\)-direction.
04

Identify Intercepts in Two-Dimensions

When \(x = 0\), \(y = 2\). When \(y = 0\), \(x = 2\). These intercepts in the \(xy\)-plane help in sketching the line of intersection with the plane \(z=0\).
05

Sketch the Plane

Start by drawing the \(xy\)-plane and plot the intercepts \( (0, 2) \) and \( (2, 0) \). The line connecting these points indicates where the plane intersects the \(xy\)-plane. From this line, draw lines parallel to the \(z\)-axis extending upwards and downwards to represent the full plane in \( \mathbb{R}^{3} \).

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Key Concepts

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

Understanding the Plane Equation in 3D Space
A plane equation in 3D space is essentially an algebraic expression used to represent a flat surface that extends infinitely in all directions. The general form of a plane's equation is given by \( ax + by + cz = d \). Each variable in this equation corresponds to one of the coordinates in 3D: \(x\), \(y\), and \(z\). The coefficients \(a\), \(b\), and \(c\) define the orientation of the plane, while \(d\) determines its position relative to the origin.

In the given problem, the equation \(x + y = 2\) indicates that we are dealing with a plane. By rewriting it as \(x + y + 0z = 2\), we can recognize it as a plane parallel to the \(z\)-axis.
Intercepts in 3D Space
Intercepts provide vital points where the plane crosses each of the axes. These points are crucial for visualizing the plane in 3D space. An intercept occurs when two of the variables equal zero, allowing the plane to intersect a particular axis.

For the equation \(x + y = 2\), let's find the intercepts:
  • **X-axis Intercept:** When \(y = 0\), the equation becomes \(x = 2\). Therefore, the intercept on the x-axis is at the point \((2, 0, 0)\).
  • **Y-axis Intercept:** Conversely, when \(x = 0\), \(y = 2\). So the intercept on the y-axis is at \((0, 2, 0)\).
Note that there is no \(z\)-axis intercept as the plane is parallel to this axis.
Understanding Z-axis Parallel Planes
Z-axis parallel planes are an intriguing concept. These are planes that do not incline or curve towards the \(z\)-axis in any way. Instead, they maintain an ongoing distance from it, never crossing or intersecting the \(z\)-axis.

In mathematical terms, this means the \(z\)-term in the equation (represented as \(cz\)) is zero. For example, in the equation \(x + y = 2\), the \(z\)-component is absent, effectively making \(c = 0\).

This characteristic indicates that every point on the plane has the same \(z\)-value, or rather, the \(z\) value does not affect the position of the plane, leading it to be infinitely extensive along the \(z\)-direction.

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Most popular questions from this chapter

\(29-36\) Reduce the equation to one of the standard forms, classify the surface, and sketch it. $$4 x^{2}+y^{2}+4 z^{2}-4 y-24 z+36=0$$

Find an equation of the plane. The plane through the point \((6,3,2)\) and perpendicular to the vector \(\langle- 2,1,5\rangle\)

Find an equation for the plane consisting of all points that are equidistant from the points \((1,0,-2)\) and \((3,4,0)\) .

Find an equation for the surface obtained by rotating the parabola \(y=x^{2}\) about the \(y\) -axis.

Use intercepts to help sketch the plane. \(2 x+5 y+z=10\)
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