Problem 45 - Algebra and Trigonometry

Question

Explain why it is impossible for a vector to have the given direction angles. $$ \alpha=20^{\circ}, \quad \beta=45^{\circ} $$

Step-by-Step Solution

Answer

The sum of squares of cosines exceeds 1, making these direction angles impossible.

Step 1: Understanding Direction Angles

Direction angles \(\alpha\), \(\beta\), and \(\gamma\) are the angles that a vector makes with the positive x, y, and z axes, respectively. For a vector to have valid direction angles, the cosine values of these angles squared should sum to 1.

Step 2: Express the Cosine of the Angles

Calculate \(\cos(\alpha)\) and \(\cos(\beta)\):\[\cos(20^{\circ}) \approx 0.9397, \quad \cos(45^{\circ}) \approx 0.7071.\]

Step 3: Use the Direction Cosines Formula

For valid direction angles, the direction cosines must satisfy the equation:\[\cos^2(\alpha) + \cos^2(\beta) + \cos^2(\gamma) = 1.\] Substitute the known values:\[0.9397^2 + 0.7071^2 + \cos^2(\gamma) = 1.\]

Step 4: Calculate the Sum of Found Values

Calculate:\[0.8821 + 0.5 = 1.3821.\] Since this is greater than 1, it is impossible to find a \(\gamma\) such that \(\cos^2(\gamma)\) makes the total equal to 1.

Step 5: Conclude the Impossibility

Since the sum of the squares of \(\cos(\alpha)\) and \(\cos(\beta)\) already exceeds 1, no value of \(\cos(\gamma)\) can make the equation true. Hence, it is impossible for a vector to have both \(\alpha = 20^{\circ}\) and \(\beta = 45^{\circ}\).

Key Concepts

Direction Cosines

Direction cosines are essential in understanding the orientation of a vector in three-dimensional space. They are the cosines of the angles that a vector makes with the positive x, y, and z axes, known as direction angles. These angles are denoted by \( \alpha \), \( \beta \), and \( \gamma \), respectively. To determine if a set of angles can be direction angles for a vector, there is a specific rule to follow: the sum of the squares of the direction cosines must equal one.

This is mathematically expressed as:

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

This equation ensures the consistency of the vector's magnitude when projected onto the coordinate axes. It also reflects the unit circle nature in three dimensions. When the sum of these squares is not equal to one, it indicates an invalid set of direction angles for any vector.

Trigonometric Identities

Trigonometric identities play a crucial role in solving problems involving direction angles. When working with direction angles, you will often encounter straightforward trigonometric functions such as sine and cosine. These identities are mathematical statements that are true for all values of the angle, and they assist us in transforming and simplifying expressions.

Key trigonometric identities include:

• \( \cos^2(\theta) + \sin^2(\theta) = 1 \)
• \( 1 + \tan^2(\theta) = \sec^2(\theta) \)

In our specific context, we primarily use \( \cos(\alpha) \) and \( \cos(\beta) \) to find admissible direction angles based on their cosine values squared. Checking these values against known sum identities helps verify whether it's possible to have a corresponding \( \gamma \). It is also through these identities that the impossibility of certain direction angles becomes evident, especially when the sum exceeds 1.

3D Vectors

3D vectors represent quantities that have both magnitude and direction in three-dimensional space. These vectors are typically expressed in the form \( \mathbf{v} = \langle a, b, c \rangle \), where \( a \), \( b \), and \( c \) are the components in the x, y, and z directions, respectively.

Understanding how vectors orient in 3D involves direction angles, which clarify the vector's inclination to each axis. For a vector defined by its direction angles, the cosine of each angle determines how closely aligned the vector is to that axis.

These angles reflect certain characteristics:

• \( \alpha \) - angle with the x-axis
• \( \beta \) - angle with the y-axis
• \( \gamma \) - angle with the z-axis

Considering the vector's geometric nature, it's clear why the direction cosines sum equation \( \cos^2(\alpha) + \cos^2(\beta) + \cos^2(\gamma) = 1 \) is necessary. It ensures the vector conforms to the unit sphere on which valid direction vectors rest. Thus, if any pair of direction angles results in a sum greater than one, like in our given problem, it's impossible to determine a third valid angle, highlighting a fundamental characteristic of vectors in 3D space.