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Chapter 11: Problem 7

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

Short Answer

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

Step by step solution

01

Identify the Given Point

The line must pass through the point \[(1,1,1)\].This point serves as the initial position or starting point for the parametric equations.
02

Determine the Direction Vector

Since the line is parallel to the \(z\)-axis, the direction vector will have components indicating movement along the \(z\)-axis and no movement along the \(x\)-axis and \(y\)-axis. The vector is\[(0, 0, 1).\]
03

Construct Parametric Equations

To form the parametric equations of the line, we use the formula\[(x, y, z) = (x_0, y_0, z_0) + t(a, b, c)\]where \((x_0, y_0, z_0) = (1, 1, 1)\) and the direction vector \((a, b, c) = (0, 0, 1)\). Substituting these values, we get:\[x = 1, \]\[y = 1, \]\[z = 1 + t.\]
04

Write the Parametric Equations

The parametric equations for the line are:- \(x = 1\)- \(y = 1\)- \(z = 1 + t\)for \(t\) in \(\mathbb{R}\). These equations show that as \(t\) varies, the point moves along the \(z\)-axis, starting from \(z = 1\).

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Key Concepts

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

Direction Vector
In parametric equations, a direction vector plays a crucial role. It tells us how a line moves through space. The direction vector is a set of components \(a, b, c\) that indicates movement in each of the x, y, and z directions.
For a line parallel to the z-axis, the direction vector will reflect movement only along this axis.

Let's break this down with an example:
  • If a line is parallel to the z-axis, it means there is no movement in the x and y directions.
  • Thus, the direction vector is \(0, 0, 1\). The numbers \(0, 0\) signify no change in x and y, while \(+1\) indicates one unit of movement up along the z-axis for each increase of parameter \(t\) by 1.
Understanding this vector is key when constructing parametric equations, as it dictates how the line advances in space from its initial point.
Parallel Lines
Parallel lines are lines in space that do not intersect, no matter how far you extend them. They have the same direction vectors, which means they follow the same path.
When we say a line is parallel to the z-axis, we're specifically indicating that its direction vector aligns with the z-axis:

Here鈥檚 what you should know:
  • Parallel lines share the same direction vector. In our context, \(0, 0, 1\).
  • Since they move only along the z-axis, their x and y values remain unchanged.
This shared direction vector assures they remain equidistant forever. The parametric equations thus ensure the x and y components are constant, reflecting their parallel nature.
Z-Axis
The z-axis is an important component of three-dimensional space, usually representing depth or height.
When considering lines parallel to the z-axis, these lines will shift vertically but not horizontally.

Here's why that matters:
  • In the equation \(z = 1 + t\), the term \(1 + t\) shows the z-axis influence. This suggests that as parameter \(t\) changes, the point advances along z.
  • Every fixed value of \(t\) gives a location on the z-axis, while \(x = 1\) and \(y = 1\) confirm no lateral movement.
Understanding how the z-axis functions in this context clarifies how these lines align in 3D space, spotlighting vertical motion and fixed lateral coordinates.

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Most popular questions from this chapter

Sketch the surfaces HYPERBOLIC PARABOLOIDS $$y^{2}-x^{2}=z$$

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Sketch the surfaces HYPERBOLOIDS $$\left(y^{2} / 4\right)-\left(x^{2} / 4\right)-z^{2}=1$$
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