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Chapter 17: Problem 9

Find parametric equations for the line. The line parallel to the \(z\) -axis passing through the point (1,0,0).

Short Answer

Expert verified
The parametric equations are \(x = 1\), \(y = 0\), \(z = t\).

Step by step solution

01

Identify the Direction Vector

Since the line is parallel to the z-axis, its direction vector is along the z-axis. This can be represented as \( \langle 0, 0, 1 \rangle \). This vector shows movement exclusively in the z-direction, without changes in the x or y directions.
02

Write the General Parametric Form

The general parametric equations for a line through a point \( (x_0, y_0, z_0) \) with direction vector \( \langle a, b, c \rangle \) are:\[x = x_0 + at, \y = y_0 + bt, \z = z_0 + ct.\]Here, \(t\) is a parameter.
03

Substitute the Point and Direction Vector

The line passes through the point \((1,0,0)\) and has the direction vector \( \langle 0, 0, 1 \rangle \). Substituting these into the parametric equations, we have:\[x = 1 + 0t, \y = 0 + 0t, \z = 0 + 1t.\]
04

Simplify the Parametric Equations

Simplifying the equations from Step 3 yields:\[x = 1, \y = 0, \z = t.\]These describe a line parallel to the z-axis passing through the point (1,0,0).

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Key Concepts

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

Direction Vector
In the context of parametric equations, a direction vector is a key concept. It defines the trajectory or path of a line in three-dimensional space. When a line is said to be parallel to an axis, its direction vector will represent movement solely in that direction.
For example, consider a line parallel to the z-axis. Its direction vector will be \( \langle 0, 0, 1 \rangle \). This vector indicates that there is no movement along the x or y-axis, only along the z-axis.
  • The first component (0) signifies no change in the x-direction.
  • The second component (0) signifies no change in the y-direction.
  • The third component (1) signifies a change in the z-direction.
Using a direction vector is crucial because it simplifies the process of finding parametric equations, guiding how the equations are constructed to describe the line's path.
Line Parallel to Axis
When discussing a line that is parallel to one of the coordinate axes, it is essential to understand how this affects its parametric equations. A line parallel to the z-axis behaves distinctively.
  • For a line to be parallel to the z-axis, it must have a direction vector like \( \langle 0, 0, 1 \rangle \).
  • This indicates that there is no deviation along the x and y axes; the line extends infinitely only in the z-direction.
Knowing that a line is parallel to an axis helps in constructing simple parametric equations where the other variables remain constant. This concept simplifies the visualization and calculation of the line's trajectory.
Z-Axis
The z-axis is one of the three axes in a 3D coordinate system, along with the x-axis and y-axis. It typically represents the vertical direction. When dealing with lines and their parametric equations, understanding the z-axis is fundamental.
  • A line parallel to the z-axis means it will only vary in the z-direction while maintaining a constant position in the x and y directions.
  • This scenario is common in problems where vertical movement is the focus, such as in this exercise.
Therefore, in a line parallel to the z-axis, the parametric equations become fairly straightforward, highlighting the critical role that the z-axis plays in describing such lines.
Parametric Form
The parametric form is a mathematical way of expressing the equations of a line as a set of equations. These equations describe a line's points based on a single parameter, often denoted as \( t \). Here鈥檚 a breakdown:
  • The general parametric equations for a line passing through a point \( (x_0, y_0, z_0) \) with a direction vector \( \langle a, b, c \rangle \) are:
  • \( x = x_0 + at \)
  • \( y = y_0 + bt \)
  • \( z = z_0 + ct \)
These formulas allow us to determine any point on the line by varying \( t \). For the specific case of a line through point \( (1,0,0) \) with direction vector \( \langle 0,0,1 \rangle \), the parametric form simplifies to \( x = 1, y = 0, \text{ and } z = t \). This straightforward representation showcases the elegant utility of the parametric form in clearly defining the position of a line in space.

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Most popular questions from this chapter

The motion of a particle is given by \(\vec{r}(t)=R \cos (\omega t) \vec{i}+\) \(R \sin (\omega t) \vec{j},\) with \(R>0, \omega>0\) (a) Show that the particle moves on a circle and find the radius, direction, and period. (b) Determine the velocity vector of the particle and its direction and speed. (c) What are the direction and magnitude of the acceleration vector of the particle?

Sketch the vector field and its flow. $$\vec{v}=2 \vec{j}$$

Give an example of: A nonconstant vector field that is parallel to \(\vec{i}+\vec{j}+\vec{k}\) at every point.

Find \(c\) so that one revolution about the \(z\) axis of the helix gives an increase of \(\Delta z\) in the \(z\) -coordinate. $$x=2 \cos t, y=2 \sin t, z=c t, \Delta z=15$$

Suppose \(\vec{r}(t)=\cos t \vec{i}+\sin t \vec{j}+2 t \vec{k}\) represents the position of a particle on a helix, where \(z\) is the height of the particle above the ground. (a) Is the particle ever moving downward? When? (b) When does the particle reach a point 10 units above the ground? (c) What is the velocity of the particle when it is 10 units above the ground? (d) When it is 10 units above the ground, the particle leaves the helix and moves along the tangent. Find parametric equations for this tangent line.
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