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Chapter 11: Problem 5

Find parametric equations for the line whose vector equation is given. $$ \begin{array}{l}{\text { (a) }\langle x, y\rangle=\langle 2,-3\rangle+ t\langle 1,-4\rangle} \\ {\text { (b) } x \mathbf{i}+y \mathbf{j}+z \mathbf{k}=\mathbf{k}+t(\mathbf{i}-\mathbf{j}+\mathbf{k})}\end{array} $$

Short Answer

Expert verified
(a) Parametric equations: \( x = 2 + t \), \( y = -3 - 4t \). (b) Parametric equations: \( x = t \), \( y = -t \), \( z = 1 + t \).

Step by step solution

01

Understand the Vector Equation (Part a)

The vector equation for part (a) is given as \( \langle x, y \rangle = \langle 2, -3 \rangle + t \langle 1, -4 \rangle \). This equation describes a line in a two-dimensional space where \( t \) is a parameter that varies over all real numbers.
02

Extract Parametric Equations (Part a)

To find the parametric equations for \( x \) and \( y \), we separate the vector equation into its components. The equation can be rewritten as \( x = 2 + t \cdot 1 \) and \( y = -3 + t \cdot (-4) \). Therefore, the parametric equations are \( x = 2 + t \) and \( y = -3 - 4t \).
03

Understand the Vector Equation (Part b)

The vector equation for part (b) is given as \( x \mathbf{i} + y \mathbf{j} + z \mathbf{k} = \mathbf{k} + t(\mathbf{i} - \mathbf{j} + \mathbf{k}) \). This line lies in three-dimensional space, and \( t \) is a parameter.
04

Extract Parametric Equations (Part b)

Express the equation in vector form as \( \langle x, y, z \rangle = \langle 0, 0, 1 \rangle + t \langle 1, -1, 1 \rangle \). By separating the components, we get the parametric equations: \( x = t \), \( y = -t \), and \( z = 1 + t \).

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Key Concepts

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

Vector Equations
Vector equations are a concise way to describe lines using vectors and a parameter, often noted as \( t \). In mathematics, vectors are used to represent direction and magnitude. They are often expressed in the form \( \langle a, b, c \rangle \), where \( a, b, \) and \( c \) are the components of the vector. In the context of vector equations:
  • A point on the line is often expressed as an initial position vector, such as \( \langle 2, -3 \rangle \) in two-dimensional space or \( \langle 0, 0, 1 \rangle \) in three-dimensional space.
  • The direction of the line is represented by another vector, such as \( \langle 1, -4 \rangle \) or \( \langle 1, -1, 1 \rangle \), which indicates how the line extends through space.
  • A parameter \( t \) allows the variable vector to run through all points on the line.
When solving vector equations, ensure you understand the role of each component and how it helps to represent a geometric line. With this understanding, it becomes easier to translate vector equations into parametric equations, which indicate each component's position on the line at any point.
Three-Dimensional Space
Three-dimensional (3D) space is a geometric setting in which three values (dimensions) are required to determine the position of a point. These dimensions are commonly represented by \( x, y, \) and \( z \) coordinates. Here are some critical insights about this space:
  • In 3D space, points are expressed as \( (x, y, z) \), whereas directions are represented by vectors such as \( x \mathbf{i} + y \mathbf{j} + z \mathbf{k} \), where \( \mathbf{i}, \mathbf{j}, \text{ and } \mathbf{k} \) are the unit vectors in the respective directions.
  • When you have a vector equation like \( \langle x, y, z \rangle = \langle 0, 0, 1 \rangle + t\langle 1, -1, 1 \rangle \), it describes a line in 3D space by indicating its starting point \( \langle 0, 0, 1 \rangle \) and the direction it travels \( \langle 1, -1, 1 \rangle \).
  • This is especially useful in fields like physics and engineering where objects move through three-dimensional environments.
Understanding how to interpret equations and vectors in \( 3D \) space is a fundamental skill in advanced math and science courses.
Two-Dimensional Space
Two-dimensional (2D) space is a simpler setting involving only two values: typically \( x \) and \( y \). This space can easily be visualized on a flat plane or graph with horizontal and vertical axes. Here are some essential aspects:
  • In 2D, any point can be represented as \( (x, y) \) and direction vectors as \( \langle a, b \rangle \). For example, a line can be represented by a vector equation like \( \langle x, y \rangle = \langle 2, -3 \rangle + t\langle 1, -4 \rangle \).
  • By converting these vector equations into parametric ones, such as \( x = 2 + t \) and \( y = -3 - 4t \), students can see how a point moves along a line in this plane as \( t \) changes.
  • This approach helps students understand linear equations and slopes in algebra, as well as vectors in physics.
Working with lines in \( 2D \) space provides a foundational understanding that prepares students for more complex multi-dimensional problems in future studies.

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Most popular questions from this chapter

An equation of a surface is given in rectangular coordinates. Find an equation of the surface in (a) cylindrical coordinates and (b) spherical coordinates. $$ x^{2}=16-z^{2} $$

Find an equation of the plane that satisfies the stated conditions. The plane that contains the line \(x=3 t, y=1+t, z=2 t\) and is parallel to the intersection of the planes \(y+z=-1\) and \(2 x-y+z=0\)

An equation is given in spherical coordinates. Express the equation in rectangular coordinates and sketch the graph. $$ \theta=\pi / 3 $$

Use a theorem from plane geometry to show that if \(\mathbf{u}\) and \(\mathbf{v}\) are vectors in 2 -space or 3 -space, then $$ \|\mathbf{u}+\mathbf{v}\| \leq\|\mathbf{u}\|+\|\mathbf{v}\| $$ which is called the triangle inequality for vectors. Give some examples to illustrate this inequality.

(a) Describe the line whose symmetric equations are $$ \frac{x-1}{2}=\frac{y+3}{4}=z-5 $$ (see Exercise 52 ). (b) Find parametric equations for the line in part (a).
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