Question

A pair of lines in \(\mathbb{R}^{3}\) are said to be skew if they are neither parallel nor intersecting. Determine whether the following pairs of lines are parallel, intersecting, or skew. If the lines intersect, determine the point(s) of intersection. \(\mathbf{r}(t)=\langle 1+6 t, 3-7 t, 2+t\rangle;\) \(\mathbf{R}(s)=\langle 10+3 s, 6+s, 14+4 s\rangle\)

Short answer

Answer: The given lines are skew lines.

Step-by-step solution

Identify the direction vectors and points

For each line, we can find the direction vector and a point on the line. Write the lines in the form \(\mathbf{r}(t)=\langle 1+6 t, 3-7 t, 2+t\rangle\) and \(\mathbf{R}(s)=\langle 10+3 s, 6+s, 14+4 s\rangle\) as \(\mathbf{r}(t) = \mathbf{a} + t\mathbf{d}\) and \(\mathbf{R}(s) = \mathbf{b} + s\mathbf{e}\). For line 1, \(\mathbf{a} = \langle 1, 3, 2 \rangle\), \(\mathbf{d} = \langle 6, -7, 1\rangle\). For line 2, \(\mathbf{b} = \langle 10, 6, 14 \rangle\), \(\mathbf{e} = \langle 3, 1, 4 \rangle\).

Check for parallel lines

To check if the lines are parallel, we need to determine if their direction vectors are proportional. If there exists a constant \(k\) such that \(\mathbf{d} = k\mathbf{e}\), then the lines are parallel. However, comparing the direction vectors, there is no constant \(k\) such that \(\langle 6, -7, 1 \rangle = k\langle 3, 1, 4\rangle\). Therefore, the lines are not parallel.

Set up the system of linear equations

Now, we need to determine if the lines intersect by solving a system of linear equations. If the points are equal, then \(\mathbf{a} + t\mathbf{d} = \mathbf{b} + s\mathbf{e}\), that is, \(1+6t = 10+3s\), \(3-7t = 6+s\), and \(2+t = 14+4s\). Write this system of linear equations as follows: 1. \(6t - 3s = 9\) 2. \(-7t - s = 3\) 3. \(t - 4s = -12\)

Solve the system of linear equations

Solve this system of linear equations by using any method (e.g., substitution, elimination). To make the calculation simpler, we eliminate \(t\) first by subtracting equation 3 from equation 2: \(-3t + 3s = 15\) Add equation 1 to the new equation: \(0 = 24\) The result is a contradiction, which tells us that the system of equations has no solution.

Draw the conclusion

Since the given lines are not parallel and there is no solution to the system of linear equations, the lines must be skew lines. Therefore, the given lines are skew lines, and they do not intersect each other.

Key concepts

Vector Calculus

Vector calculus is a field of mathematics that focuses on vector functions and their derivatives or integrals. In relation to skew lines in 3D space, vector calculus simplifies the understanding of lines as vector functions. These functions describe lines by incorporating both a direction vector and a position vector.

In the given exercise, the vector functions \(\mathbf{r}(t)\) and \(\mathbf{R}(s)\) represent two lines in three-dimensional space. The role of vector calculus here is vital, as it provides the tools needed to analyze the lines' behaviors—whether they intersect, are parallel, or skew. By utilizing vector addition and scalar multiplication, we can easily navigate complex 3D structures in a mathematical way.

Direction Vectors

Direction vectors are vectors that indicate the direction in which a line extends in space. In our exercise, the direction vectors are identified for each line: \(\mathbf{d} = \langle 6, -7, 1 \rangle\) and \(\mathbf{e} = \langle 3, 1, 4 \rangle\). Knowing these vectors is crucial, as they help determine the relationship between lines.

If the direction vectors are proportional, the lines are parallel, as they would extend in the same direction. However, if no scalar multiple can relate the direction vectors, the lines cannot be parallel. To check for intersection, we make use of the direction vectors by setting up a system of linear equations, reflecting the assumption that two points on each line coincide at some parameter values for \(t\) and \(s\).

System of Linear Equations

A system of linear equations consists of two or more linear equations with the same set of variables. In 3D space, we use these systems to determine if two lines intersect. The system includes three separate equations, one for each component (x, y, z) of the vector equation.

In our exercise, the system was derived from the assumption of intersection. This system of equations can often be solved using methods such as substitution or elimination. When the system has no solution, it means the lines do not intersect and, thus, if they are not parallel, they are skew. The steps provided in the solution used the elimination method to deduce the lack of a solution, confirming the lines are skew.

Line Intersection in 3D

Exploring the intersection of two lines in 3D requires an analysis of their vector equations. These equations express a line's path in terms of parametric variables, like \(t\) or \(s\), allowing us to navigate along their length. For lines to intersect, their respective positions need to match at some points along their paths.

The system of linear equations formed by equating the vector functions at potential intersection points captures this condition. If the system has a unique solution, it yields the intersection point's coordinates. Conversely, an inconsistent system, which emerges in the case of the lines in the exercise, suggests no such point exists, indicating that the lines are skew.