Question

Find the largest and smallest values of the sum of the acute angles that a line through the origin makes with the three coordinate axes.

Short answer

Maximum: 180 degrees, Minimum: 164.31 degrees

Step-by-step solution

Understand the Problem

A line through the origin makes angles \(\theta_x\), \(\theta_y\), and \(\theta_z\) with the x, y, and z-axes respectively. The goal is to find the maximum and minimum values of the sum of these angles.

Use the Direction Cosines

For angles \(\theta_x\), \(\theta_y\), and \(\theta_z\), the direction cosines are \(\cos(\theta_x)\), \(\cos(\theta_y)\), and \(\cos(\theta_z)\). For a line through the origin, the relationship \(\cos^2(\theta_x) + \cos^2(\theta_y) + \cos^2(\theta_z) = 1\) holds.

Find the Sum of the Angles

To find the sum of \(\theta_x\), \(\theta_y\), and \(\theta_z\), let's analyze the potential extreme values. We know \(\theta_x\), \(\theta_y\), and \(\theta_z\) are acute angles, meaning each is between 0 and 90 degrees.

Analyze Extreme Values

If the line coincides with one of the axes, say the x-axis, then \(\theta_x = 0\) and \(\theta_y = 90\) and \(\theta_z = 90\). Sum \(\theta_x + \theta_y + \theta_z = 0 + 90 + 90 = 180 \) degrees. This is the maximum value.

Evaluate Minimum Sum

If the line is equally inclined to all three axes, \(\cos(\theta_x) = \cos(\theta_y) = \cos(\theta_z) = \frac{1}{\sqrt{3}}\). Then \(\theta_x = \theta_y = \theta_z = \arccos(\frac{1}{\sqrt{3}})\approx 54.77 \) degrees. Therefore, the sum is \(3 \cdot \arccos(\frac{1}{\sqrt{3}})\approx 164.31 \) degrees. This is the minimum value.

Key concepts

Understanding Acute Angles

An acute angle is any angle that is less than 90 degrees but greater than 0 degrees. In geometry, these angles are often encountered, especially when studying the properties of triangles and lines in relation to coordinate axes. Acute angles are fundamental when analyzing the inclination of a line to coordinate axes. They ensure that the sum of angles provides meaningful and realistic interpretations in three-dimensional space. When considering a line through the origin, the objective is to find the sum of these acute angles with respect to the x, y, and z-axes.

Coordinate Axes in Three-Dimensional Space

The coordinate axes, namely the x, y, and z-axes, form the basis of three-dimensional space. Each axis represents a dimension, and they intersect at the origin (0,0,0). When a line passes through the origin, it forms angles with these axes. These angles are crucial in understanding the spatial orientation of the line. The direction cosines, represented by \(\backslash cos(\theta_x)\), \(\backslash cos(\theta_y)\), and \(\backslash cos(\theta_z)\), describe the cosine of the angles the line makes with each respective axis.
For any line through the origin, the relationship \(\backslash cos^2(\theta_x) + \backslash cos^2(\theta_y) + \backslash cos^2(\theta_z) = 1\) holds true. This equation is a key aspect of direction cosines and helps in determining the angles made by the line with each axis. Understanding this relationship is essential for analyzing the line's orientation in three-dimensional space.

Identifying Extreme Values of Angles

Extreme values refer to the maximum and minimum sums of the angles that the line makes with the coordinate axes. To find these values, consider different scenarios for the angles.

• Maximum Sum: The maximum value occurs when the line aligns with one of the coordinate axes fully. For example, if the line coincides with the x-axis, the angles with the y and z-axes will be 90 degrees each, resulting in a sum of 180 degrees.
• Minimum Sum: The minimum value occurs when the line is equally inclined to all three axes. Here, the angles are such that \(\cos(\theta_x) = \cos(\theta_y) = \cos(\theta_z) = \frac{1}{\backslash sqrt{3}}\). The angle for each axis is approximately 54.77 degrees, leading to a sum of around 164.31 degrees.

By evaluating these extreme scenarios, students can better understand how the orientation of the line affects the sum of the angles with respect to the coordinate axes.