Problem 54 - Calculus for Biology and Medicine

Question

Use the following steps to show that if two nonvertical lines \(l_{1}\) and \(l_{2}\) with slopes \(m_{1}\) and \(m_{2}\), respectively, are perpendicular, then \(m_{1} m_{2}=-1\) : Assume that \(m_{1}<0\) and \(m_{2}>0\). (a) Use a graph to show that if \(\theta_{1}\) and \(\theta_{2}\) are the respective angles of inclination of the lines \(l_{1}\) and \(l_{2}\), then \(\theta_{1}=\theta_{2}+\frac{\pi}{2} .\) (The angle of inclination of a line is the angle \(\theta \in[0, \pi)\) between the line and the positively directed \(x\) -axis.) (b) Use the fact that \(\tan (\pi-x)=-\tan x\) to show that \(m_{1}=\) \(\tan \theta_{1}\) and \(m_{2}=\tan \theta_{2}\) (c) Use the fact that \(\tan \left(\frac{\pi}{2}-x\right)=\cot x\) and \(\cot (-x)=-\cot x\) to show that \(m_{1}=-\cot \theta_{2}\) (d) From the latter equation, deduce the truth of the claim set forth at the beginning of this problem.

Step-by-Step Solution

Answer

If two nonvertical lines are perpendicular, then the product of their slopes is -1.

Step 1: Understand Line Inclinations

For two lines to be perpendicular, the angle between them should be \(\frac{\pi}{2}\) radians. When nonvertical line \(l_1\) with slope \m_1\ and line \(l_2\) with slope \m_2\ are perpendicular, the angle of inclination \(\theta_1\) of \(l_1\) is related to \(\theta_2\) of \(l_2\) by \(\theta_1 = \theta_2 + \frac{\pi}{2}\).

Step 2: Analyze Slope and Angle Relationship

The slope of a line is given by the tangent of its angle of inclination. Therefore, \( m_1 = \tan \theta_1 \) and \( m_2 = \tan \theta_2 \). By knowing the tangent function's periodicity, \( \tan(\pi - x) = -\tan x \,\) it's important that \( m_1 < 0 \) and \( m_2 > 0 \) since one is negative and the other positive.

Step 3: Use Angle Transformation

Given \(\theta_1 = \theta_2 + \frac{\pi}{2}\), we use the identity \ \tan \left(\frac{\pi}{2} - x\right) = \cot x\. Hence, \( m_1 = \tan(\theta_2 + \frac{\pi}{2}) = -\cot(\theta_2)\), leading to \( m_1 = -\cot( \theta_2 )\).

Step 4: Establish Relation Between Slopes

Since \(m_1 = -\cot(\theta_2)\) and \(m_2 = \tan(\theta_2)\), then \(-\cot(\theta_2) = -\frac{1}{\tan(\theta_2)} \), thus establishing that the product of the slopes is: \( m_1 m_2 = (-\cot \theta_2)(\tan \theta_2) = -1 \).

Key Concepts

Slope

When we talk about the slope of a line, we're discussing a measure of how steep the line is. It tells us how much the line rises or falls as we move along it horizontally. The slope is usually represented by the letter \( m \).

• A positive slope means the line is moving upwards as you go from left to right.
• A negative slope indicates the line is moving downwards.
• If the slope is zero, the line is perfectly horizontal.
• For a vertical line, the slope is undefined.

The slope of a line is calculated as the change in the \( y \)-value divided by the change in the \( x \)-value between two points on the line. Mathematically, if you have two points \((x_1, y_1)\) and \((x_2, y_2)\), the slope is given by: \\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
For two lines to be perpendicular, the product of their slopes must equal \(-1\). This occurs because perpendicular lines have slopes that are negative reciprocals of each other.

Angle of Inclination

The angle of inclination of a line is the angle that the line makes with the positive \( x \)-axis. It is measured in the plane from the positive direction of the \( x \)-axis, counterclockwise to the direction of the line. This angle is usually denoted by \( \theta \) and can be anywhere from \( 0 \) to \( \pi \) radians.
When two lines are perpendicular, the angle of inclination between them is \( \frac{\pi}{2} \) radians, or 90 degrees. If you have a line with an angle of inclination \( \theta_1 \) and another line with \( \theta_2 \), and they are perpendicular, the relationship is: \\[ \theta_1 = \theta_2 + \frac{\pi}{2} \]
This relationship helps in finding the slopes of the lines since each slope can be associated with the tangent of these inclination angles. Integration of these angles into the function of the slope provides a geometric interpretation essential in trigonometry.

Tangent Function

The tangent function is a fundamental concept in trigonometry. It relates an angle in a right triangle to the ratio of the lengths of the opposite side to the adjacent side. For an angle \( \theta \), \( \tan \theta \) is defined as:
\[ \tan \theta = \frac{\text{opposite}}{\text{adjacent}} \]
In the context of line slopes, the tangent function is used to connect the angle of inclination of a line to its slope. Specifically, the slope \( m \) of a line is equal to the tangent of its angle of inclination \( \theta \): \[ m = \tan \theta \]
One key property of the tangent function is its periodicity, meaning it repeats its values in regular intervals. The periodicity property used here is that \( \tan(\pi - x) = -\tan x \). This helps establish how the signs of the slopes change under different angular conditions, such as with perpendicular lines.

Cotangent Function

The cotangent function is the reciprocal of the tangent function. For an angle \( \theta \), the cotangent is defined as the ratio of the adjacent side to the opposite side in a right triangle, or: \\[ \cot \theta = \frac{\text{adjacent}}{\text{opposite}} = \frac{1}{\tan \theta} \]
In terms of slopes of lines, the cotangent is particularly useful when assessing perpendicular lines. Since these slopes are negative reciprocals of each other, understanding cotangent allows us to express one slope in terms of another's cotangent:\[ m_1 = -\cot( \theta_2 ) \]
This equation helps transition from the idea of slopes to using trigonometric identities. The periodicity of the cotangent, especially the identity \( \cot(-x) = -\cot x \), is crucial in transforming the angles and understanding their extensions and relations.