Problem 13
Question
Find parametrizations for the line segments joining the points in Exercises \(13-20 .\) Draw coordinate axes and sketch each segment, indicating the direction of increasing \(t\) for your parametrization. $$ (0,0,0), \quad(1,1,3 / 2) $$
Step-by-Step Solution
Verified Answer
The parametrization is \((x(t), y(t), z(t)) = (t, t, \frac{3}{2}t)\) for \(0 \leq t \leq 1\).
1Step 1: Understand the Points Given
The given points are \((0, 0, 0)\) and \((1, 1, \frac{3}{2})\). These are the endpoints of the line segment we need to parametrize.
2Step 2: Determine the Vector Between Points
Create a vector from the first point to the second point. The vector is calculated by subtracting the coordinates of the first point from the second: \((1 - 0, 1 - 0, \frac{3}{2} - 0) = (1, 1, \frac{3}{2})\). This vector gives the direction of the line segment.
3Step 3: Write the Parametrization
The parametrization of the line segment can be written as \((x,y,z)(t) = (0, 0, 0) + t(1, 1, \frac{3}{2})\). Simplifying, we have: \(x(t) = t, y(t) = t, z(t) = \frac{3}{2}t\), where \(0 \leq t \leq 1\).
4Step 4: Sketch the Line Segment
Draw a 3D coordinate system. Plot the points \((0, 0, 0)\) and \((1, 1, \frac{3}{2})\). Draw a line segment between these points, indicating the direction from \(t = 0\) to \(t = 1\) along the segment.
Key Concepts
Vector Mathematics3D Coordinate GeometryParametric EquationsLine Segments in Three-Dimensional Space
Vector Mathematics
Vectors are fundamental in mathematics and physics. They have both a direction and a magnitude. We often represent them as arrows in space. Vectors can describe a variety of entities, such as forces and velocities. In our case, they help describe the direction and distance from one point to another in space.
When working with vectors in 3D, we usually deal with three components, often denoted as \((x, y, z)\). You can think of these components as how far along the x, y, and z axes the vector reaches. The vector in our problem, \((1, 1, \frac{3}{2})\), points from the origin to the point \((1, 1, \frac{3}{2})\).
When working with vectors in 3D, we usually deal with three components, often denoted as \((x, y, z)\). You can think of these components as how far along the x, y, and z axes the vector reaches. The vector in our problem, \((1, 1, \frac{3}{2})\), points from the origin to the point \((1, 1, \frac{3}{2})\).
- To find the vector between two points, you subtract the coordinates of the first point from those of the second.
- The vector gives us a direction for the line.
- Magnitude is given by the vector length: \( \sqrt{x^2 + y^2 + z^2} \).
3D Coordinate Geometry
The three-dimensional coordinate system is used to depict points, lines, and planes in space. It extends the 2D concept by adding an additional axis. Each point is defined by three coordinates \((x, y, z)\).
In 3D geometry, we often want to visualize them by plotting points and drawing lines or surfaces. When working with line segments, each segment is defined by its endpoints, like our points \((0, 0, 0)\) and \((1, 1, \frac{3}{2})\).
In 3D geometry, we often want to visualize them by plotting points and drawing lines or surfaces. When working with line segments, each segment is defined by its endpoints, like our points \((0, 0, 0)\) and \((1, 1, \frac{3}{2})\).
- The x-axis runs along the horizontal direction.
- The y-axis runs vertically.
- The z-axis typically represents depth (imagine it coming towards you).
Parametric Equations
Parametric equations offer a way to describe a geometric object, like a line, by using parameters. Instead of expressing \(y\) solely in terms of \(x\), we consider each coordinate as a function of one or more parameters.
In the case of a line segment in 3D, we often use a parameter \(t\) that varies from 0 to 1. Each value of \(t\) produces a point on the line segment. This is shown as:
- \(x(t) = t\)
- \(y(t) = t\)
- \(z(t) = \frac{3}{2}t\)
As \(t\) moves from 0 to 1, the point moves from the start to the endpoint. Choosing parametric equations makes calculations simpler, especially when dealing with complex curves or moving objects in animations.
In the case of a line segment in 3D, we often use a parameter \(t\) that varies from 0 to 1. Each value of \(t\) produces a point on the line segment. This is shown as:
- \(x(t) = t\)
- \(y(t) = t\)
- \(z(t) = \frac{3}{2}t\)
As \(t\) moves from 0 to 1, the point moves from the start to the endpoint. Choosing parametric equations makes calculations simpler, especially when dealing with complex curves or moving objects in animations.
Line Segments in Three-Dimensional Space
A line segment in 3D is simply a straight path connecting two points in space. Unlike a continuous line extending infinitely, a segment has defined start and end.
To represent a line segment, we employ vectors and parametric equations. These help define its position and length. In our exercise, the end points are \((0, 0, 0)\) and \((1, 1, \frac{3}{2})\).
To represent a line segment, we employ vectors and parametric equations. These help define its position and length. In our exercise, the end points are \((0, 0, 0)\) and \((1, 1, \frac{3}{2})\).
- The vector \((1, 1, \frac{3}{2})\) gives the direction of the segment.
- Parametric equations such as \(x(t) = t\), \(y(t) = t\), and \(z(t) = \frac{3}{2}t\) help us trace the path.
- The limitations \(0 \leq t \leq 1\) ensure the path stays within the segment boundaries.
Other exercises in this chapter
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