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Chapter 12: Problem 13

Find parametrizations for the line segments joining the points. Draw coordinate axes and sketch each segment, indicating the direction of increasing \(t\) for your parametrization. \begin{equation}(0,0,0), \quad(1,1,3 / 2)\end{equation}

Short Answer

Expert verified
The parametrization is \\( x(t) = t \\), \\( y(t) = t \\), \\( z(t) = \frac{3}{2}t \\) for \\( 0 \leq t \leq 1 \\."

Step by step solution

01

Identify Line Segment

The line segment joins the points \( (0,0,0) \) and \( (1,1,\frac{3}{2}) \). This is a 3-dimensional vector from the origin to the point \( (1,1,\frac{3}{2}) \).
02

Determine Direction Vector

To find the direction vector of the line segment, subtract the coordinates of the starting point from the ending point. The direction vector is \( \langle 1-0, 1-0, \frac{3}{2}-0 \rangle = \langle 1, 1, \frac{3}{2} \rangle \).
03

Write Parametric Equations

The parametric equations for a line segment can be written as: \[\begin{align*}x(t) &= x_0 + at \y(t) &= y_0 + bt \z(t) &= z_0 + ct\end{align*}\]Substitute \( (x_0, y_0, z_0) = (0,0,0) \) and direction vector components \( (a,b,c) = (1,1,\frac{3}{2}) \):\[\begin{align*}x(t) &= t \y(t) &= t \z(t) &= \frac{3}{2}t\end{align*}\]where \( 0 \leq t \leq 1 \).
04

Sketch and Direction Indication

Draw a 3D coordinate system and mark the points \( (0,0,0) \) and \( (1,1,\frac{3}{2}) \). Sketch the line segment connecting these points. To indicate the direction, draw arrows along the segment, pointing from \( (0,0,0) \) to \( (1,1,\frac{3}{2}) \). This shows the direction of increasing \( t \).

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Key Concepts

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

line segment parametrization
Line segment parametrization is a method used to represent the path between two distinct points in space using a parameter, usually denoted as \( t \). This parameter allows us to express the position along the line segment as a continuous transition from the starting point to the end point.
  • The basic idea is to interpret the segment as being traced out as \( t \) moves from 0 to 1.
  • The point where \( t = 0 \) corresponds to the starting point, and \( t = 1 \) corresponds to the ending point.
  • In this exercise, the parametric form is expressed as \( x(t) = x_0 + at \), \( y(t) = y_0 + bt \), and \( z(t) = z_0 + ct \), where \((x_0, y_0, z_0)\) is the initial point and \((a, b, c)\) is the direction vector.
Using parametrization helps in analyzing movements or paths in various dimensions, especially within 3D spaces.
3D coordinate system
In a 3D coordinate system, every point is described by a triplet of numbers \((x, y, z)\). This system provides a way to locate points in a three-dimensional space.
  • Think of it like using coordinates on a map, but with an added dimension for height or depth.
  • The origin, labeled as \((0, 0, 0)\), is the central point where all three spatial axes intersect.
  • Each axis is perpendicular to the others, providing a unique reference framework to plot any point in the space.
For this exercise, understanding how to navigate and sketch in a 3D coordinate system is crucial as the points like \((0,0,0)\) and \((1,1,\frac{3}{2})\) are plotted, and a line segment is drawn to connect them.
direction vector
The direction vector is a fundamental concept when working with line segments in any dimension. It describes the orientation and magnitude from the starting point to the end point of the line segment.
  • This vector is obtained by subtracting the initial point from the final point, giving us the components \((a, b, c)\).
  • In our example, the direction vector is calculated as \( \langle 1, 1, \frac{3}{2} \rangle \),
  • This tells us that moving from the initial point to the final point involves moving 1 unit in the x-direction, 1 unit in the y-direction, and \(\frac{3}{2}\) units in the z-direction.
Direction vectors are crucial for defining the path of a line segment in a precise mathematical form.
vector subtraction
Vector subtraction is a mathematical process that helps in finding the direction vector. It involves subtracting the respective components of one vector from another.
  • This operation enables the calculation of the direction vector between two points.
  • In our exercise, the subtraction \((1-0, 1-0, \frac{3}{2}-0)\) is performed to find the difference in each coordinate direction.
  • The resulting vector \( \langle 1, 1, \frac{3}{2} \rangle \) shows how the line progresses from the point \((0,0,0)\) to the point \((1,1,\frac{3}{2})\).
Understanding vector subtraction allows for precise descriptions of lines, helping you manipulate and understand spatial relationships in geometry.

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Most popular questions from this chapter

Find the areas of the parallelograms whose vertices are given in Exercises \(35 - 40 .\) $$ A ( - 1,2 ) , \quad B ( 2,0 ) , \quad C ( 7,1 ) , \quad D ( 4,3 ) $$

In Exercises \(55 - 57 ,\) determine whether the given points are coplanar. $$A ( 0,0,4 ) , \quad B ( 6,2,0 ) , \quad C ( 2 , - 1,1 ) , \quad D ( - 3 , - 4,3 )$$

Find the distance from the \begin{equation} \begin{array}{l}{\text { a. } x \text { -axis to the plane } z=3 .} \\ {\text { b. origin to the plane } 2=z-x \text { . }} \\ {\text { c. point }(0,4,0) \text { to the plane } y=x \text { . }}\end{array} \end{equation}

Triangle area Find a \(2 \times 2\) determinant formula for the area of the triangle in the \(x y\) -plane with vertices at \(( 0,0 ) , \left( a _ { 1 } , a _ { 2 } \right) ,\) and \(\left( b _ { 1 } , b _ { 2 } \right)\) . Explain your work.

Find all points that simultaneously lie 3 units from each of the points \((2,0,0),(0,2,0),\) and \((0,0,2) .\)
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