Problem 11 - Essential Calculus Early Transcendentals

Question

Is the line through \((-4,-6,1)\) and \((-2,0,-3)\) parallel to the line through \((10,18,4)\) and \((5,3,14) ?\)

Step-by-Step Solution

Answer

Yes, the lines are parallel because their direction vectors are scalar multiples of each other.

Step 1: Find Direction Vector for First Line

To determine if two lines are parallel, we start by finding the direction vector of the first line. To do this, subtract the coordinates of the first point from the second point on the line. So, for the line through points \((-4,-6,1)\) and \((-2,0,-3)\), the direction vector is calculated as follows:\[(-2 - (-4), 0 - (-6), -3 - 1) = (2, 6, -4)\]Hence, the direction vector for the first line is \( (2, 6, -4) \).

Step 2: Find Direction Vector for Second Line

Next, find the direction vector of the line through points \( (10,18,4) \) and \( (5,3,14) \) by subtracting the first point from the second point:\[(5 - 10, 3 - 18, 14 - 4) = (-5, -15, 10)\]Thus, the direction vector for the second line is \((-5, -15, 10)\).

Step 3: Compare the Direction Vectors

Determine if the direction vectors of the two lines are scalar multiples of each other, which would indicate that the lines are parallel. We compare the direction vectors \( (2, 6, -4) \) and \( (-5, -15, 10) \). Check if there exists a constant \kappa\ such that:\[2 = -5\kappa, \quad 6 = -15\kappa, \quad -4 = 10\kappa\]Solving these equations:\[kappa = -\frac{2}{5}, \quad kappa = -\frac{6}{15} = -\frac{2}{5}, \quad kappa = -\frac{4}{10} = -\frac{2}{5}\]All ratios are the same, indicating the vectors are scalar multiples and the lines are parallel.

Key Concepts

Direction Vectors in Vector Calculus

Direction vectors provide critical insights into the characteristics of a line, especially when working in vector calculus. A direction vector essentially gives us the direction in which a line extends. To find it, you subtract the components of one endpoint of a line segment from the components of the other endpoint.

For example, consider a line passing through the points \((-4, -6, 1)\) and \((-2, 0, -3)\). To find the direction vector, you perform the following computation: \[-2 - (-4), \quad 0 - (-6), \quad -3 - 1 = (2, 6, -4)\].

The resultant direction vector is \(2, 6, -4\), pointing from the first point toward the second. Using this method, each line in space can be described uniquely by its direction vector.

Understanding Parallel Lines

Parallel lines in three-dimensional space are lines that never meet, regardless of how far they are extended. In vector calculus, two lines are parallel if their direction vectors are scalar multiples of each other.

Let's consider two lines, one defined by a direction vector \(2, 6, -4\) and the other by \(-5, -15, 10\). To determine if these lines are parallel, we check if there exists a scalar \(\kappa\) such that one vector is a multiple of the other:

• 2 = -5\(\kappa\)
• 6 = -15\(\kappa\)
• -4 = 10\(\kappa\)

Solving these gives us \( \kappa = -\frac{2}{5}\).

Since \( \kappa \) is consistent across all components, the lines are indeed parallel. This insight is vital for understanding and analyzing line relationships in various fields, such as physics and engineering.

Scalar Multiples and Their Role

Scalar multiples are fundamental in determining if two vectors are parallel. A vector \((a, b, c)\) is a scalar multiple of another vector \((d, e, f)\) if there exists a constant \( \kappa \) such that \( a = \kappa d, \quad b = \kappa e, \quad c = \kappa f\).

In our example, the direction vectors \(2, 6, -4\) and \(-5, -15, 10\) are scalar multiples because \( \kappa = -\frac{2}{5}\) satisfies all component equations. This calculation confirms that two vectors, when scaled by the same factor, maintain their parallel relationship.

The concept of scalar multiples is not only applicable to vectors, it permeates all linear spaces, facilitating the solving of equations and the understanding of vector relationships.