Chapter 10: Problem 18
Determine if the described lines are the same line, parallel lines, intersecting or skew lines. If intersecting, give the point of intersection. $$ \begin{array}{l} \overrightarrow{\ell_{1}}(t)=\langle 1,1,1\rangle+t\langle 3,1,3\rangle \\ \vec{\ell}_{2}(t)=\langle 7,3,7\rangle+t\langle 6,2,6\rangle . \end{array} $$
Short Answer
Expert verified
The lines are parallel.
Step by step solution
01
Identify Direction Vectors
The direction vector for \( \ell_1 \) is \( \langle 3, 1, 3 \rangle \) and the direction vector for \( \ell_2 \) is \( \langle 6, 2, 6 \rangle \). Determine if these direction vectors are scalar multiples of each other to test if the lines are parallel or the same.
02
Check Parallel or Same Line
Divide the components of \( \ell_2 \) by \( \ell_1 \): \( \frac{6}{3} = 2 \), \( \frac{2}{1} = 2 \), \( \frac{6}{3} = 2 \). Since the ratio is constant, the direction vectors are parallel.
03
Check if Lines Coincide
Next, check if a scalar multiply of the direction vector will give you a point on the other line. Substitute a point from \( \ell_1 \), say \( \langle 1,1,1 \rangle \), into the parametric equation of \( \ell_2 \). If valid, lines are the same; if not, they are only parallel.
04
Test Coincidence
Check if there exists \( t \) such that \( \langle 7, 3, 7 \rangle + t \langle 6, 2, 6 \rangle = \langle 1, 1, 1 \rangle + s\langle 3, 1, 3 \rangle \) is true for some \( s \). Set the equations equal and solve: \( s \langle 3, 1, 3 \rangle = \langle -6, -2, -6 \rangle - t \langle 6, 2, 6 \rangle \). No solution exists.
Unlock Step-by-Step Solutions & Ace Your Exams!
-
Full Textbook Solutions
Get detailed explanations and key concepts
-
Unlimited Al creation
Al flashcards, explanations, exams and more...
-
Ads-free access
To over 500 millions flashcards
-
Money-back guarantee
We refund you if you fail your exam.
Over 30 million students worldwide already upgrade their learning with Vaia!
Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Direction Vectors
Direction vectors are crucial for understanding the orientation of lines in space. They help us determine whether two lines are parallel, perpendicular, or skew. A direction vector is essentially a vector that represents the direction along which a line extends. When given the parametric equations of a line, the direction vector can be found as the coefficients of the parameter.
For example, with the line \( \overrightarrow{\ell_{1}}(t)=\langle 1,1,1\rangle+t\langle 3,1,3\rangle \), the direction vector is \( \langle 3,1,3 \rangle \). It tells us that as the parameter \( t \) increases, how the line moves in 3D space. Similarly, for another line \( \vec{\ell}_{2}(t)=\langle 7,3,7\rangle+t\langle 6,2,6\rangle \), the direction vector is \( \langle 6,2,6 \rangle \).
To determine if two lines are parallel, we compare their direction vectors. If one vector is a scalar multiple of the other, the lines are parallel. Scalar multiplication of vectors is straightforward; you multiply each component of the vector by the scalar. This comparison helps us conclude if the lines maintain a consistent directionality across space.
For example, with the line \( \overrightarrow{\ell_{1}}(t)=\langle 1,1,1\rangle+t\langle 3,1,3\rangle \), the direction vector is \( \langle 3,1,3 \rangle \). It tells us that as the parameter \( t \) increases, how the line moves in 3D space. Similarly, for another line \( \vec{\ell}_{2}(t)=\langle 7,3,7\rangle+t\langle 6,2,6\rangle \), the direction vector is \( \langle 6,2,6 \rangle \).
To determine if two lines are parallel, we compare their direction vectors. If one vector is a scalar multiple of the other, the lines are parallel. Scalar multiplication of vectors is straightforward; you multiply each component of the vector by the scalar. This comparison helps us conclude if the lines maintain a consistent directionality across space.
Parametric Equations
Parametric equations are a way to express the position of a point on a line using a parameter, typically denoted as \( t \). They offer a flexible method for describing lines in both two-dimensional and three-dimensional spaces. Each line in a parametric form is expressed as a combination of a point and a direction vector.
In the given example, \( \overrightarrow{\ell_{1}}(t)=\langle 1,1,1\rangle+t\langle 3,1,3\rangle \) states that any point on line \( \ell_1 \) can be found by starting at the point \( \langle 1,1,1 \rangle \) and extending in the direction of \( \langle 3,1,3 \rangle \) by a factor of \( t \). This formulation is powerful because it allows varying \( t \) to trace every point on the line.
Similarly, \( \vec{\ell}_{2}(t)=\langle 7,3,7\rangle+t\langle 6,2,6\rangle \) describes line \( \ell_2 \). This equation also comprises a point \( \langle 7,3,7 \rangle \) and a direction vector. By understanding parametric equations, one can easily navigate through the spatial positioning of lines and identify intersections and parallels.
In the given example, \( \overrightarrow{\ell_{1}}(t)=\langle 1,1,1\rangle+t\langle 3,1,3\rangle \) states that any point on line \( \ell_1 \) can be found by starting at the point \( \langle 1,1,1 \rangle \) and extending in the direction of \( \langle 3,1,3 \rangle \) by a factor of \( t \). This formulation is powerful because it allows varying \( t \) to trace every point on the line.
Similarly, \( \vec{\ell}_{2}(t)=\langle 7,3,7\rangle+t\langle 6,2,6\rangle \) describes line \( \ell_2 \). This equation also comprises a point \( \langle 7,3,7 \rangle \) and a direction vector. By understanding parametric equations, one can easily navigate through the spatial positioning of lines and identify intersections and parallels.
Parallel Lines
Parallel lines maintain the same direction without ever intersecting, unless they are coincident (the same line in space). The primary way to check if two lines are parallel using their parametric equations is to compare the direction vectors. If the direction vector of one line is a scalar multiple of the other, they are parallel.
In the described problem, the direction vector of \( \ell_1 \) is \( \langle 3,1,3\rangle \) and that of \( \ell_2 \) is \( \langle 6,2,6 \rangle \). By checking the components of each direction vector, we observe that each component of \( \ell_2 \) is twice that of \( \ell_1 \), confirming the ratio is constant: \( \frac{6}{3} = 2 \), \( \frac{2}{1} = 2 \), \( \frac{6}{3} = 2 \). Hence, \( \ell_1 \) and \( \ell_2 \) are parallel.
Even though they are parallel, it's important to verify they are not coincident by matching points on one line to those on the other using their parametric equations. In this instance, no point \( t \) satisfies both parametric equations simultaneously, confirming they are indeed merely parallel and not the same line.
In the described problem, the direction vector of \( \ell_1 \) is \( \langle 3,1,3\rangle \) and that of \( \ell_2 \) is \( \langle 6,2,6 \rangle \). By checking the components of each direction vector, we observe that each component of \( \ell_2 \) is twice that of \( \ell_1 \), confirming the ratio is constant: \( \frac{6}{3} = 2 \), \( \frac{2}{1} = 2 \), \( \frac{6}{3} = 2 \). Hence, \( \ell_1 \) and \( \ell_2 \) are parallel.
Even though they are parallel, it's important to verify they are not coincident by matching points on one line to those on the other using their parametric equations. In this instance, no point \( t \) satisfies both parametric equations simultaneously, confirming they are indeed merely parallel and not the same line.
Line Intersection
Line intersection occurs when two lines in space cross at a single point. To find if and where two lines intersect, you must equate their parametric equations and solve for the parameter values. If a solution exists, the lines intersect at a specific point. Otherwise, they are parallel or skew.
For the lines \( \overrightarrow{\ell_{1}}(t)=\langle 1,1,1\rangle+t\langle 3,1,3\rangle \) and \( \vec{\ell}_{2}(t)=\langle 7,3,7\rangle+t\langle 6,2,6\rangle \), we theoretically set up an equation where both lines equal the same 3D point. However, when attempting to solve \( \langle 7, 3, 7 \rangle + t \langle 6, 2, 6 \rangle = \langle 1, 1, 1 \rangle + s \langle 3, 1, 3 \rangle \), no solution for \( s \) and \( t \) makes this equation true for all components.
This outcome confirms that the lines do not intersect at any point in space, supporting the conclusion that they are parallel, as determined by their respective direction vectors.
For the lines \( \overrightarrow{\ell_{1}}(t)=\langle 1,1,1\rangle+t\langle 3,1,3\rangle \) and \( \vec{\ell}_{2}(t)=\langle 7,3,7\rangle+t\langle 6,2,6\rangle \), we theoretically set up an equation where both lines equal the same 3D point. However, when attempting to solve \( \langle 7, 3, 7 \rangle + t \langle 6, 2, 6 \rangle = \langle 1, 1, 1 \rangle + s \langle 3, 1, 3 \rangle \), no solution for \( s \) and \( t \) makes this equation true for all components.
This outcome confirms that the lines do not intersect at any point in space, supporting the conclusion that they are parallel, as determined by their respective direction vectors.
