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Magazine Chapter 1: Problem 37
Require some knowledge of chemical notation. Find a line that lies entirely in the set defined by the cquation \(x^{2}+y^{2}-z^{2}=1\).
Short Answer
Expert verified
A line is \( (x,y,z) = (1,0,t) \) for any real number \( t \).
Step by step solution
01
Understand the Equation
The given equation is \(x^2 + y^2 - z^2 = 1\). This represents a hyperboloid in three-dimensional space. We need to find a line that lies entirely within this hyperboloid.
02
Parametrize the Line
Consider a line parametrized by \(\mathbf{r}(t) = (x_0 + at, y_0 + bt, z_0 + ct)\), where \(x_0, y_0, z_0\) is a point on the line and \(a, b, c\) are the direction ratios of the line.
03
Insert Parametrization into Equation
Substitute \(x(t) = x_0 + at\), \(y(t) = y_0 + bt\), and \(z(t) = z_0 + ct\) into the equation \(x^2 + y^2 - z^2 = 1\). You get: \((x_0 + at)^2 + (y_0 + bt)^2 - (z_0 + ct)^2 = 1\).
04
Simplify and Solve for Line Conditions
Simplify the equation to find conditions on \(x_0, y_0, z_0, a, b,\) and \(c\). Choose \(a = 0, b = 0\) and \(c = 1\) for simplification, leading to \((x_0)^2 + (y_0)^2 - (z_0 + t)^2 = 1\).
05
Find Conditions for Parameters
For a constant solution independent of \(t\), we should have \(z_0 = 0\). The simplified equation \(x_0^2 + y_0^2 - t^2 = 1\) implies a suitable constant pair for \(x_0,y_0\) needs to be a circle of radius 1 when \(t = 0\). Choose advantageously \(x_0 = 1, y_0 = 0\).
06
Determine the Line
Analyze how the choice leads to \(\mathbf{r}(t) = (1,0,t)\), that will satisfy the original equation because \((1)^2 + (0)^2 - (t)^2 = 1 - t^2\) holds always for small \(t\) under continuity.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Understanding Hyperboloids
A hyperboloid is a type of surface found in three-dimensional space. It is defined by a specific type of quadratic equation, like the one we see in the exercise: \( x^2 + y^2 - z^2 = 1 \). This equation forms what is known as a hyperboloid of one sheet. Imagine this as a three-dimensional, hourglass-shaped surface.
Hyperboloids are important because they can model real-world objects and structures, such as cooling towers in nuclear power plants.
Here's some key aspects about hyperboloids:
Hyperboloids are important because they can model real-world objects and structures, such as cooling towers in nuclear power plants.
Here's some key aspects about hyperboloids:
- They are ruled surfaces, meaning they can be made up of straight lines.
- They have two types: one-sheet and two-sheet hyperboloids based on the sign of the equation.
- The cross-sections parallel to the coordinate planes can be ellipses or hyperbolas.
What Are Parametric Equations?
Parametric equations allow us to describe a curve or a line using a single variable, commonly denoted as \( t \). They are particularly useful in vector calculus for modeling paths in space.
For example, a line can be described by the parametric equations \(x(t) = x_0 + at\), \(y(t) = y_0 + bt\), \(z(t) = z_0 + ct\). Here, \(x_0, y_0, z_0\) represents a starting point, and \(a, b, c\) are the direction numbers of the line.
Key points about parametric equations:
For example, a line can be described by the parametric equations \(x(t) = x_0 + at\), \(y(t) = y_0 + bt\), \(z(t) = z_0 + ct\). Here, \(x_0, y_0, z_0\) represents a starting point, and \(a, b, c\) are the direction numbers of the line.
Key points about parametric equations:
- They can capture more complex curves and paths than simple functions of \(x\) or \(y\).
- They are crucial for describing lines that lie entirely within surfaces like hyperboloids.
- Parametric forms are widely used in physics and engineering for trajectory modeling.
Navigating Three-Dimensional Space
Three-dimensional space is a mathematical environment where points are defined by three coordinates: \(x, y, z\). These coordinates help describe positions in a volume, as opposed to a flat, two-dimensional space.
In vector calculus, we often navigate this space using vectors and equations to describe complex surfaces like hyperboloids.
Let's summarize 3D space attributes:
In vector calculus, we often navigate this space using vectors and equations to describe complex surfaces like hyperboloids.
Let's summarize 3D space attributes:
- Three axes: the x-axis (horizontal), y-axis (depth), and z-axis (vertical).
- Allows modeling of real-world objects more accurately than two dimensions.
- Surfaces like planes, cylinders, and hyperboloids are defined using 3D equations.
