Problem 4 Find parametric equations for th... [FREE SOLUTION]

Chapter 11: Problem 4

Find parametric equations for the lines. The line through \(P(1,2,0)\) and \(Q(1,1,-1)\)

Short Answer

Expert verified

The parametric equations are \(x = 1\), \(y = 2 - t\), \(z = -t\).

Step by step solution

Determine the Direction Vector

First, we need to find the direction vector of the line that passes through points \(P(1, 2, 0)\) and \(Q(1, 1, -1)\). The direction vector \( \mathbf{v} \) can be found by subtracting the coordinates of \(P\) from \(Q\). Thus, \( \mathbf{v} = (1-1, 1-2, -1-0) = (0, -1, -1) \).

Write the Parametric Equations

With the direction vector \( \mathbf{v} = (0, -1, -1) \) and a point on the line, say \(P(1, 2, 0)\), we can write the parametric equations of the line. If \((x, y, z)\) are the coordinates on the line, then the parametric equations are: \( x = 1 \), \( y = 2 - t \), \( z = -t \), where \( t \) is the parameter.

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Key Concepts

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

Direction Vector

The direction vector is fundamental when dealing with parametric equations of a line. Essentially, it tells us the direction in which the line extends.
To find the direction vector for a line through two points, we subtract the coordinates of the starting point from the ending point. For example:

Given two points, \( P(1, 2, 0) \) and \( Q(1, 1, -1) \), the direction vector \( \mathbf{v} \) is calculated as \( (1-1, 1-2, -1-0) = (0, -1, -1) \).
This vector \( \mathbf{v} \) represents the line's slope in a three-dimensional space.
Having a zero in any component indicates no change along that axis; in this case, the line remains constant in the x-direction.

This concept allows us to visualize how a line moves through space, and it's a stepping stone for creating parametric equations.

Point on a Line

At the heart of defining a line parametrically is identifying a known point through which the line passes. This point allows us to anchor our parametric equations, providing a starting reference.
In the given problem, the point \( P(1, 2, 0) \) serves this purpose.
Using this known point, we configure our equations:

The x-coordinate 1 in \( P \) is directly used in the equation \( x = 1 \).
The other coordinates serve as the basis for additional calculations involving the parameter \( t \), which represents how far we move along our direction vector.

Having at least one point is crucial, as it ensures that our line is fixed in space rather than simply floating directionally.

Line through Points

Creating parametric equations for a line that spans two points relies heavily on combining the direction vector and a known point. We find the direction vector by computing the difference between the two points as shown earlier.
After determining the direction vector \( \mathbf{v} = (0, -1, -1) \), we utilize the known point \( P(1, 2, 0) \), and express the line with a parameter \( t \).

Such parametric line equations are in the form: \( x = x_0 + t \cdot a \), \( y = y_0 + t \cdot b \), and \( z = z_0 + t \cdot c \).
Here, \( (a, b, c) \) are the components of the direction vector, and \( (x_0, y_0, z_0) \) are the coordinates of the initial point.
This results in the parametric equations: \( x = 1 \), \( y = 2 - t \), \( z = -t \).

These equations beautifully combine direction and position, artfully expressing the line in terms of a parameter \( t \), which simplifies solving and analyzing the line's path.

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91影视

Find parametric equations for the lines. The line through \(P(1,2,0)\) and \(Q(1,1,-1)\)

Short Answer

Step by step solution

Determine the Direction Vector

Write the Parametric Equations

Key Concepts

Direction Vector

Point on a Line

Line through Points

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