Problem 4 - Algebra and Trigonometry Real Mathematics, Real People

Question

What is the relationship between two lines whose slopes are -3 and \(\frac{1}{3} ?\)

Step-by-Step Solution

Answer

The given lines are perpendicular.

Step 1: Identify the slopes

The two given slopes are -3 and \(\frac{1}{3}\).

Step 2: Determine if the lines are parallel

The lines would be parallel if both slopes were equal. However, -3 and \(\frac{1}{3}\) are not equal so the lines are not parallel.

Step 3: Determine if the lines are perpendicular

Two lines are perpendicular to each other if the product of their slopes is -1. In this case, the product of -3 and \(\frac{1}{3}\) is -1. Thus, these two lines are perpendicular.

Key Concepts

Perpendicular Lines

When we talk about lines being perpendicular, we refer to two lines that intersect at a right angle (90 degrees). A fundamental characteristic of perpendicular lines is that the product of their slopes is -1. This relationship is crucial for quickly determining the orientation of two lines in a plane.

For instance, in the textbook exercise, the two slopes given are -3 and \( \frac{1}{3} \). To figure out if these lines are perpendicular, we multiply the slopes together. If the result is -1, then we can confirm that they are indeed perpendicular. Here's the quick calculation:

\( (-3) \times \frac{1}{3} = -1 \)

Through this calculation, we established that the two lines are perpendicular. This is an important concept in geometry and is widely applicable in fields that require precise angle measurements, such as engineering, architecture, and even various art forms.

Parallel Lines

Parallel lines are lines in a plane that never meet; they have the same slope and are always the same distance apart. In the provided exercise, the concept of parallelism is tested. The key factor to recognize parallel lines is that their slopes must be exactly equal.

Let's consider an example. If we have two lines with slopes of 2 and 2, or -1/2 and -1/2, they would be parallel because their slopes match. If the slopes do not equal each other, as in the case of the slopes -3 and \( \frac{1}{3} \), we determine that the lines are not parallel. As you proceed further in mathematics, understanding the properties of parallel lines can be useful in coordinate geometry, trigonometry, and calculus.

Slope of a Line

The slope of a line is a measure of how steep a line is and the direction in which it tilts. It's calculated by taking the ratio of the vertical change to the horizontal change between two points on a line. The formula for the slope, 'm', between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by:

\( m = \frac{(y_2 - y_1)}{(x_2 - x_1)} \)

In the context of our exercise, the slopes provided are -3 and \( \frac{1}{3} \). The slope -3 implies a line that falls steeply from left to right, while \( \frac{1}{3} \) indicates a much gentler rise from left to right. Understanding the slope is critical for later topics in algebra, calculus, and physics, as it is often a foundational element in problem-solving processes involving linear relationships.