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A direction vector is essential for understanding the orientation of a line in 3-dimensional space. In parametric equations of lines, the direction vector helps us

• Determine the line's slope along each axis.
• Understand how the line extends through the space.

For instance, the line \(L_1\) is given by the equation \(x=2t, y=0, z=3, t \in \mathbb{R}\). We can deduce its direction vector, \(\langle 2, 0, 0 \rangle\), by examining the coefficients of the parameter \(t\). This direction vector signifies that the line progresses infinitely along the x-axis but remains static in the y and z directions.

Similarly, for the line \(L_2: x=0, y=8+s, z=7+s, s \in \mathbb{R}\), its direction vector is \(\langle 0, 1, 1 \rangle\). This direction vector shows that \(L_2\) does not move along the x-axis but stretches equally in the positive y and z directions.

To determine relationships between lines, we compare their direction vectors. If they are proportional, the lines might be parallel or equal. A lack of proportionality, like between \(\langle 2, 0, 0 \rangle\) and \(\langle 0, 1, 1 \rangle\), tells us that the lines cannot be equal or parallel.

Parametric equations provide a way to describe a line's path in 3D space using a parameter, say \(t\) or \(s\). Each component of the line is expressed in terms of the parameter, illustrating how each spatial coordinate changes as the parameter varies.

For example, the parametric equations of the line \(L_1\) are:

• \(x = 2t\)
• \(y = 0\)
• \(z = 3\)

These equations show that altering the parameter \(t\) changes the x-coordinate, while y and z remain constant. Therefore, \(L_1\) is a straight line parallel to the x-axis.

In contrast, the line \(L_2\) is described as:

• \(x = 0\)
• \(y = 8 + s\)
• \(z = 7 + s\)

Here, adjusting \(s\) increases both y and z, while x stays at zero, indicating \(L_2\) runs diagonally in the yz-plane. Understanding parametric equations helps determine line direction and spatial behavior.

The intersection of lines happens when they cross each other at a specific point. To find an intersection, you typically set the parametric equations of one line equal to the corresponding equations of another and solve for the parameter values that satisfy these conditions.

For lines \(L_1\) and \(L_2\):- Set \(2t = 0\)- \(0 = 8+s\)- \(3 = 7+s\)Solving these equations gives a unique challenge here because the only consistent value for \(t\) is 0, while the values for \(s\) are inconsistent (-8 and -4). The mismatch indicates that no single \(t\), \(s\) pair satisfies all equations simultaneously.

Thus, \(L_1\) and \(L_2\) do not cross each other, confirming they do not intersect. Understanding when and why lines intersect or don't can simplify geometrical reasoning.

Parallel lines are an important concept in geometry. They never meet, running alongside each other indefinitely. For lines to be parallel, their direction vectors must be proportional. If one vector can be obtained by multiplying another by a scalar, then those lines share the same direction and will never meet.

Given \(L_1\) with direction vector \(\langle 2, 0, 0 \rangle\) and \(L_2\) with \(\langle 0, 1, 1 \rangle\), we notice they aren't parallel. The vectors are different in every component, meaning there isn't a scalar that can transform one vector into another.

This indicates that \(L_1\) and \(L_2\) traverse entirely different paths in space and can never run parallel or be parts of the same line. Recognizing parallel lines by analyzing their direction vectors helps simplify spatial relationships.