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Chapter 11: Problem 5

Interpret the graph of \(x=1\) in the contexts of (a) a number line (b) 2 -space (c) 3 -space.

Short Answer

Expert verified
Single point on the number line, vertical line in 2D, plane in 3D.

Step by step solution

01

Interpret on a Number Line

On a number line, the graph of \(x=1\) is represented by a single point. This point is located at the position corresponding to the number 1 on the horizontal line, which is usually depicted as an infinite line extending in both the negative and positive directions with zero at the origin.
02

Interpret in 2-Space

In a 2-dimensional space, the graph of \(x=1\) is a vertical line. This line is parallel to the y-axis and passes through all points where the x-coordinate is equal to 1. Thus, it extends infinitely in both the positive and negative directions along the y-axis.
03

Interpret in 3-Space

In a 3-dimensional space, the graph of \(x=1\) is a plane. This plane is parallel to both the y-z plane and extends infinitely in the directions of the y and z axes. Every point on this plane has an x-coordinate of 1, while the y and z coordinates can vary freely.

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Key Concepts

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

Number Line
The number line is a straightforward way to visualize real numbers. Imagine a straight, horizontal line extending endlessly in both directions. At the center is zero, acting as a reference point. Numbers to the right of zero are positive, and those to the left are negative. When we talk about the graph of \( x = 1 \) on a number line, we are referring to a single point. This point is situated at the coordinate that marks the value of 1.
  • This means that \( x = 1 \) is a very specific location.
  • No other values are included, making it distinct and singular.
Understanding this helps clarify that an equation like \( x = 1 \) is telling us precisely one value where \( x \) resides.
2D Space
In a 2D space, represented as a flat plane, things become a little more complex than on a number line. Here, we deal with two axes 鈥� typically the x-axis running horizontally and the y-axis running vertically.When we interpret \( x = 1 \) within this space, we're looking at a vertical line.
  • This line passes through every point where the x-coordinate is exactly 1.
  • Think of it as an infinite line that stretches both upwards and downwards.
The important element here is that no matter where you move on this line, the x value doesn鈥檛 change. It remains fixed at 1, while the y-coordinates vary freely, covering all possibilities.
3D Space
When imagining 3D space, think about the world around you. It includes three axes: x, y, and z. This allows us to represent points in a more complex way by using three coordinates.For \( x = 1 \) in this context, we observe not a line but a plane. This plane is parallel to the y-z plane and extends infinitely.
  • A plane in 3D space includes all points where the x-coordinate is a constant value of 1.
  • The y and z coordinates can alter across the plane, showing infinite combinations.
This visualization highlights how, in 3D, such equations encompass vast areas, showing that a simple change in dimension alters our perspective significantly.

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Most popular questions from this chapter

Identify the surface and make a rough sketch that shows its position and orientation. $$ z=(x+2)^{2}+(y-3)^{2}-9 $$

Let \(L_{1}\) and \(L_{2}\) be the lines whose parametric equations are $$ \begin{array}{lll} L_{1}: x=1+2 t, & y=2-t, & z=4-2 t \\ L_{2}: x=9+t, & y=5+3 t, & z=-4-t \end{array} $$ (a) Show that \(L_{1}\) and \(L_{2}\) intersect at the point \((7,-1,-2)\). (b) Find, to the nearest degree, the acute angle between \(L_{1}\) and \(L_{2}\) at their intersection. (c) Find parametric equations for the line that is perpendicular to \(L_{1}\) and \(L_{2}\) and passes through their point of intersection.

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Sketch the surface. $$ z=\sqrt{1-x^{2}-y^{2}} $$

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