Problem 42 - Precalculus Mathematics for Calculus

Question

Two direction angles of a vector are given. Find the third direction angle, given that it is either obtuse or acute as indicated. (In Exercises 43 and \(44,\) round your answers to the nearest degree.) $$\beta=\frac{2 \pi}{3}, \quad \gamma=\frac{\pi}{4} ; \quad \alpha \text { is acute }$$

Step-by-Step Solution

Answer

The acute angle \( \alpha \) is 60 degrees.

Step 1: Understanding Direction Angles

For a vector in three-dimensional space, the direction angles \( \alpha, \beta, \gamma \) are measured from the axes, with \( \alpha \) measured from the x-axis, \( \beta \) from the y-axis, and \( \gamma \) from the z-axis. They relate to the direction cosines, which sum up to 1: \( \cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = 1 \).

Step 2: Convert Given Angles to Radians

The problem specifies \( \beta = \frac{2\pi}{3} \) and \( \gamma = \frac{\pi}{4} \). These are already in radians, so we can proceed to find their direction cosines.

Step 3: Calculate Direction Cosines

Calculate \( \cos \beta \) and \( \cos \gamma \): 1. \( \cos \beta = \cos\left(\frac{2\pi}{3}\right) = -\frac{1}{2} \) 2. \( \cos \gamma = \cos\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2} \)

Step 4: Apply the Direction Cosines Formula

Utilize the formula: \[ \cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = 1 \] Substitute the calculated cosines: \[ \cos^2 \alpha + \left(-\frac{1}{2}\right)^2 + \left(\frac{\sqrt{2}}{2}\right)^2 = 1 \] This simplifies to: \[ \cos^2 \alpha + \frac{1}{4} + \frac{2}{4} = 1 \] Thus, \[ \cos^2 \alpha = 1 - \frac{1}{4} - \frac{1}{2} = \frac{1}{4} \]

Step 5: Solve for \( \alpha \) Given It is Acute

Since \( \alpha \) is acute, \( \cos \alpha = \sqrt{\frac{1}{4}} = \frac{1}{2} \). The angle \( \alpha \) corresponding to \( \cos \alpha = \frac{1}{2} \) is \( \alpha = \frac{\pi}{3} \) or 60 degrees in acute form.

Key Concepts

Direction Cosines

Direction cosines play a crucial role when dealing with three-dimensional vectors.
When we talk about direction angles, it refers to the angles a vector makes with each axis in a 3D space. For a vector, these angles are typically denoted as \( \alpha, \beta, \text{ and } \gamma \), corresponding to the x, y, and z axes respectively.

The direction cosines, which are \( \cos \alpha, \cos \beta, \text{ and } \cos \gamma \), are essentially the cosines of each of these respective angles. They're important because they help define the vector's orientation in space.

Here's a key relationship to remember:

• \( \cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = 1 \)

This equation ensures that the vector maintains its directionality in three-dimensional space. When solving problems involving vectors, calculating these direction cosines becomes essential to understand how the vector is situated relative to each of the three axes.

Acute Angle

An acute angle is any angle that's less than 90 degrees.
When it comes to vector direction angles, the classification of an angle as acute can greatly influence the final results of calculations.

In our context, if we find that the cosine of a certain direction angle is positive, the angle itself will be acute. Why? Because cosine values are positive in the first quadrant, which corresponds to angles between 0 and 90 degrees.
Let's break this down:

• \( \cos \alpha = \frac{1}{2} \) implies \( \alpha = \frac{\pi}{3} \), which translates to 60 degrees.

Since the problem specifies \( \alpha \) as acute, it confirms that the angle is within this desired range (0 to 90 degrees). This understanding is critical, especially when paired with the vector's other angles that may be in various portions of the coordinate system.

Three-Dimensional Vectors

Three-dimensional vectors, often represented as \([x, y, z]\), are fundamental in describing quantities that have both a magnitude and a direction in 3D space.
They're widely used in physics, engineering, and mathematics to model real-world dimensions.

Here's what makes them special:

• They can express movement or force in any spatial direction.
• Their components (x, y, z) denote the vector's influence along each respective axis.

The direction angles \( \alpha, \beta, \gamma \) are used to determine how a vector tilts against the standard axes.
These angles help in finding how a vector is oriented in space.

For instance, a vector perpendicular to the x-axis would have a direction angle of 90 degrees with it. Knowing these angles and their cosines allows us to analyze a vector's behavior and its interactions with other vectors or forces.