Problem 15 - APEX Calculus

Question

In Exercises 15-22, determine if the described lines are the same line, parallel lines, intersecting or skew lines. If intersecting, give the point of intersection. $$ \underline{\phantom{xxx}} \begin{aligned} \vec{\ell}_{1}(t) &=\langle 1,2,1\rangle+t\langle 2,-1,1\rangle \\ \vec{\ell}_{2}(t) &=\langle 3,3,3\rangle+t\langle-4,2,-2\rangle \end{aligned} $$

Step-by-Step Solution

Answer

The lines are parallel, not the same line.

Step 1: Compare Direction Vectors

First, identify the direction vectors of the two lines: \( \mathbf{v}_1 = \langle 2, -1, 1 \rangle \) for \( \vec{\ell}_1 \) and \( \mathbf{v}_2 = \langle -4, 2, -2 \rangle \) for \( \vec{\ell}_2 \). These vectors are multiples, \( \mathbf{v}_2 = -2 \times \mathbf{v}_1 \), so the lines are parallel.

Step 2: Check for Same Line

Even though the lines are parallel, we must check if they have a point in common to determine if they are the same line. Set the equations of the lines equal to each other and solve for \( t \) and \( s \): \( \langle 1+2t, 2-t, 1+t \rangle = \langle 3-4s, 3+2s, 3-2s \rangle \).

Step 3: Solve System of Equations

From \( 1 + 2t = 3 - 4s \), we get \( 2t + 4s = 2 \). From \( 2 - t = 3 + 2s \), we get \( t + 2s = -1 \). From \( 1 + t = 3 - 2s \), we get \( t + 2s = 2 \). This inconsistency in equations means no \( t \) and \( s \) satisfy all simultaneously, so the lines are not the same.

Key Concepts

Direction Vectors

In 3D analytic geometry, direction vectors are essential for understanding the behavior of lines. They dictate the direction in which a line extends. When given a line in vector form, such as \( \vec{\ell}(t) = \mathbf{a} + t \mathbf{v} \), \( \mathbf{v} \) is the direction vector. This vector indicates how the line moves through space as the parameter \( t \) changes.
For example, in the problem at hand, we have two lines: one with a direction vector \( \langle 2, -1, 1 \rangle \) and the other with \( \langle -4, 2, -2 \rangle \).

To determine if two vectors are pointing in the same direction, and thus if the lines they define are parallel, we check if one direction vector is a scalar multiple of the other. In this case, \( \mathbf{v}_2 = -2 \times \mathbf{v}_1 \), confirming that the lines are indeed parallel. This means that the two lines will never intersect nor approach each other as \( t \) changes.

Parallel Lines

Parallel lines are a fascinating topic in 3D geometry. They can be thought of as lines that maintain a constant distance from each other in all dimensions. Two lines that are parallel in 3D space have direction vectors that are scalar multiples of each other.
This implies that they "move" through space in the same direction, even though they may be situated at different locations. While such lines never intersect, they provide valuable insights into the spatial arrangement.
In our present example, once we realize that the direction vectors \( \mathbf{v}_1 = \langle 2, -1, 1 \rangle \) and \( \mathbf{v}_2 = \langle -4, 2, -2 \rangle \) are multiples of one another, we conclude that the lines are parallel. It's crucial to ascertain whether they are the same line by checking if they share at least one point. This involves solving a system of equations to look for any shared solution for \( t \) or any parameter between the equations of the lines.

System of Equations

When determining whether two parallel lines are actually the same line, or perhaps intersect at some point, solving a system of equations is often necessary. This procedure involves setting the parametric equations of the lines equal and solving for the parameters or variables involved.
In our exercise, the line equations are \( \langle 1+2t, 2-t, 1+t \rangle \) and \( \langle 3-4s, 3+2s, 3-2s \rangle \). By equating these, we obtain a system of equations:

• \( 2t + 4s = 2 \)
• \( t + 2s = -1 \)
• \( t + 2s = 2 \)

Upon analyzing these, we discover an inconsistency, which indicates no values of \( t \) and \( s \) can simultaneously satisfy all three equations. This inconsistency demonstrates that, even though the lines are parallel, they are not coincident and do not define the same line in space. Solving such equations requires careful attention to detail and understanding the underlying geometric principles.