Question

Show that two lines in the plane with slopes \(m_{1}\) and \(m_{2}\) are perpendicular if and only if \(m_{1} m_{2}=-1 .[\) Hint : Example \(4.1 .11 .]\)

Short answer

Two lines are perpendicular if and only if the slopes multiply to -1.

Step-by-step solution

Understanding Perpendicular Lines

Two lines are perpendicular if they intersect at a right angle (90 degrees). In terms of slopes, when the product of their slopes is -1, the lines are perpendicular.

Define Slopes for Lines

Let the slope of the first line be denoted by \(m_1\), and the slope of the second line by \(m_2\). To show these lines are perpendicular, we need to check the relationship between \(m_1\) and \(m_2\).

Set Up the Product of the Slopes

Set up the equation as \(m_1 \cdot m_2 = -1\). This is the condition we aim to satisfy or verify for the lines to be perpendicular.

Combine Slopes with the Perpendicular Relation

For lines to be perpendicular, their slopes meet the criterion: \(m_1 \cdot m_2 = -1\). If this equation holds true, then it implies that the lines intersect at a right angle.

Verify with Example Calculation

From the hints, we know that in previous examples, two lines with slopes that multiply to \( -1 \) are perpendicular. Plugging in any potential values, if \( m_1 = 2 \) and \( m_2 = -\frac{1}{2} \), their product is \( 2 \times -\frac{1}{2} = -1\), confirming they are perpendicular.

Confirm Logic and Criterion

Conclude that if and only if the product of the slopes \( m_1 \) and \( m_2 \) is -1, the given lines are assuredly perpendicular. This aligns with the initial definition of perpendicular lines via angle intersections.

Key concepts

Slopes of Lines

The slope of a line is a measure of its steepness and direction. It is calculated as the ratio of the vertical change to the horizontal change between two points on the line. Mathematically, the slope (usually denoted as \( m \)) is given by the formula: \( m = \frac{y_2 - y_1}{x_2 - x_1} \). Here, \( (x_1, y_1) \) and \( (x_2, y_2) \) are coordinates of any two points on the line.

Understanding slopes is essential for analyzing relationships between lines. A larger absolute value of the slope indicates a steeper line, while a smaller value suggests a more gradual slope. If the slope is positive, the line rises from left to right; if negative, it falls from left to right. A zero slope corresponds to a horizontal line, and an undefined slope is associated with a vertical line. Slopes play a critical role in determining not just direction and steepness, but also interactions between lines, such as intersections.

Right Angle Intersection

When lines intersect at a right angle, meaning 90 degrees, they are said to be perpendicular. This type of intersection is crucial in geometry as it forms the basis of defining several geometric properties and shapes, such as squares and rectangles.

Perpendicular lines demonstrate a special relationship. They create a perfect 'corner' when they meet, resembling an 'L' shape. In real-world terms, the intersection looks like a crossroad forming four right angles. Mathematically, perpendicular lines have a specific slope condition: the product of their slopes is -1, reflecting their perfect orthogonal intersection on a plane.

Recognizing when lines intersect at right angles aids in constructing perpendicular bisectors, managing Cartesian coordinates, and solving problems related to angles and area. This foundational concept is central in many areas, from architecture to coordinate geometry.

Product of Slopes

The product of slopes is a key concept when analyzing whether two lines are perpendicular. When two lines intersect at a right angle, the slopes of these lines, when multiplied together, equal -1. This is an algebraic representation of their perpendicular relationship.

Let's use an example to illustrate this. Consider two lines with slopes \( m_1 = 2 \), and \( m_2 = -\frac{1}{2} \). Calculating the product, we have \( m_1 \cdot m_2 = 2 \times -\frac{1}{2} = -1 \). This result confirms the perpendicular nature of the lines. It's crucial to note that if this product does not equal -1, the lines do not intersect at a right angle.

This condition, \( m_1 \cdot m_2 = -1 \), is not only a mathematical criterion but also a verification tool. It helps to confirm or refute the idea of perpendicularity without directly measuring angles, streamlining the process particularly in complex mathematical tasks.