Problem 15 - Calculus: Early Transcendentals

Question

Determine whether the lines \(\langle 1,1,2\rangle+t\langle 1,2,-3\rangle\) and \(\langle 2,3,-1\rangle+t\langle 2,4,-6\rangle\) are parallel, intersect, or neither.

Step-by-Step Solution

Answer

The lines are parallel.

Step 1: Identify Direction Vectors

Examine the direction vectors of the two lines given in vector form.The direction vector for line 1 is \( \langle 1, 2, -3 \rangle \).The direction vector for line 2 is \( \langle 2, 4, -6 \rangle \).

Step 2: Check for Parallelism

To determine if the lines are parallel, check if one direction vector is a scalar multiple of the other.By comparing, \( \langle 2, 4, -6 \rangle = 2 \times \langle 1, 2, -3 \rangle \).Since the direction vectors are scalar multiples of each other, the lines are parallel.

Step 3 (Optional): Check for Coincidence

Since the lines are parallel, check if they are the same line by seeing if a point on one line lies on the other. Substitute a point from line 1 into the equation of line 2.The point \( \langle 1, 1, 2 \rangle \) doesn't satisfy the equation of line 2, therefore the lines are not coincident.

Step 4: Conclusion

Given the analysis, the lines are parallel but not coincident as they don't share any points.

Key Concepts

Vector Algebra

Vector algebra is a fundamental concept in mathematics that helps in understanding and representing geometric transformations and positions of objects in space. In the context of lines, vectors play a crucial role in defining the direction and position of the line. A vector is essentially an entity with both magnitude and direction. When dealing with line equations, we often denote lines in vector form using a direction vector and a point on the line.

Here is how it looks: a line in vector form can be written as

• \( \textbf{r} = \textbf{a} + t\textbf{b} \)

where \( \textbf{r} \) is any point on the line, \( \textbf{a} \) is a fixed point on the line, \( t \) is a scalar parameter, and \( \textbf{b} \) is the direction vector.

This form makes it easy to compute operations like determining if lines are parallel, intersect, or coincide by examining their vectors.

Parallel Lines

Parallel lines are lines that run in the same direction but never intersect. In vector algebra, determining if lines are parallel involves looking at their direction vectors. If one vector is a scalar multiple of the other, the lines are parallel.

For the exercise in question, we look at the direction vectors:

• Line 1: \( \langle 1, 2, -3 \rangle \)
• Line 2: \( \langle 2, 4, -6 \rangle \)

Notice that the direction vector of line 2 is exactly twice that of line 1, meaning

• \( \langle 2, 4, -6 \rangle = 2 \times \langle 1, 2, -3 \rangle \)

Since one direction vector is a scalar multiple of the other, this means that the two lines are parallel. Parallel lines can be very close yet never touch each other, extending infinitely in both directions.

Direction Vectors

Direction vectors are key components in describing lines in space. They provide the 'direction' part of a line's equation when expressed in vector form. Essentially, the direction vector adjusts the line's orientation and extends it along the path that the vector points to.

In the given lines:

• Direction vector for line 1 is \( \langle 1, 2, -3 \rangle \)
• Direction vector for line 2 is \( \langle 2, 4, -6 \rangle \)

The importance of direction vectors lies in their ability to help determine relationships between lines. For example, if we find that one vector is a scalar multiple of another, we know the lines are parallel. This exercise demonstrates that property well. Additionally, direction vectors are essential in intersecting and determining geometric properties of line pairs.

So, always observe the behavior of direction vectors when analyzing lines in vector form, as they are the anchor point for understanding how lines relate to one another spatially.