Problem 40 - Precalculus Mathematics for Calculus

Question

Find the direction angles of the given vector, rounded to the nearest degree. $$(2,-1,2)$$

Step-by-Step Solution

Answer

The direction angles are approximately \(48^\circ\), \(110^\circ\), and \(48^\circ\).

Step 1: Understand the concept of direction angles

Direction angles are the angles formed by a vector with the positive axes of the coordinate system. For a vector \(\mathbf{v} = (a, b, c)\), these angles \(\alpha, \beta,\) and \(\gamma\) are determined from: \(\cos \alpha = \frac{a}{||\mathbf{v}||}\), \(\cos \beta = \frac{b}{||\mathbf{v}||}\), \(\cos \gamma = \frac{c}{||\mathbf{v}||}\). We need to find these angles for the vector \((2, -1, 2)\).

Step 2: Calculate the magnitude of the vector

The magnitude of a vector \(\mathbf{v} = (a, b, c)\) is given by \(||\mathbf{v}|| = \sqrt{a^2 + b^2 + c^2}\).For our vector \((2, -1, 2)\):\[||\mathbf{v}|| = \sqrt{2^2 + (-1)^2 + 2^2} = \sqrt{4 + 1 + 4} = \sqrt{9} = 3.\]

Step 3: Find the cosine of each direction angle

Now calculate each cosine value: - \(\cos \alpha = \frac{2}{3}\)- \(\cos \beta = \frac{-1}{3}\)- \(\cos \gamma = \frac{2}{3}\)

Step 4: Calculate the direction angles

To find the direction angles, use the inverse cosine function (\( \cos^{-1} \)):- \(\alpha = \cos^{-1}\left(\frac{2}{3}\right)\)- \(\beta = \cos^{-1}\left(\frac{-1}{3}\right)\)- \(\gamma = \cos^{-1}\left(\frac{2}{3}\right)\)Calculating these gives: \(\alpha \approx 48.2^\circ\), \(\beta \approx 109.5^\circ\), \(\gamma \approx 48.2^\circ\).

Step 5: Round the direction angles to the nearest degree

Round the calculated angles to the nearest whole degree as specified:- \(\alpha \approx 48^\circ\)- \(\beta \approx 110^\circ\)- \(\gamma \approx 48^\circ\)

Key Concepts

Magnitude of a Vector

The magnitude of a vector is a crucial concept in vector mathematics, especially in 3D space. It describes the length or size of the vector, giving us a single positive value that represents the vector's "strength." To find the magnitude of any vector \(\mathbf{v} = (a, b, c)\), you use the formula: \[||\mathbf{v}|| = \sqrt{a^2 + b^2 + c^2}.\]This is an extension of the Pythagorean theorem into three dimensions, adding up the squares of the components and taking the square root of the sum.

The unit's consistency across all vector magnitudes is another intriguing detail. It doesn’t matter the direction the vector points; magnitude solely depends on its components. In our example, the vector is \(2, -1, 2\), and its magnitude is \(|\mathbf{v}| = \sqrt{4 + 1 + 4} = \sqrt{9} = 3\). Seeing your vector's magnitude as the hypotenuse of a triangle in 3D space can be a helpful way to conceptualize it.

Cosine Inverse

The inverse of a cosine function, often written as \(\cos^{-1}\), is the mathematical operation that allows us to find the angle when its cosine value is known. This is essential in calculating direction angles. Direction angles tell us how a vector is oriented relative to the coordinate axes in 3D space.

In our specific problem, once you calculate the cosine values for each direction (i.e., \(\alpha, \beta, \gamma\)), you use the inverse cosine function to determine the actual angles. Using the provided example, we found:

• \(\cos \alpha = \frac{2}{3}\),
• \(\cos \beta = \frac{-1}{3}\),
• \(\cos \gamma = \frac{2}{3}\).

To retain clarity, note that when using \(\cos^{-1}\), you must work in radians or degrees, depending on your preference, although degrees are typically more intuitive for most students.

The key here is understanding that: \(\alpha = \cos^{-1}\left(\frac{2}{3}\right), \beta = \cos^{-1}\left(\frac{-1}{3}\right), \gamma = \cos^{-1}\left(\frac{2}{3}\right)\). This process directly yields the angular measurements, providing us the vector's directional insight.

Vector in 3D

A vector in 3D space can be identified using three values: the x, y, and z components. This gives vectors the ability to represent anything from forces to velocities in three-dimensional applications, making them crucial for fields like physics, engineering, and computer graphics.

The essence of the direction angles problem is to understand how this specific vector \(\mathbf{v} = (2, -1, 2)\) aligns with the coordinate axes. Each component tells part of this story:

• The "2" in x implies a strong influence in the positive x-direction.
• The "-1" in y indicates a lesser movement in the negative y-direction.
• The second "2" in z suggests another strong pull in the positive z-direction.

By visualizing or plotting this vector, you can "see" how these measurements translate to three-dimensional space.

Understanding vectors in 3D is about combining both numeric (algebraic) and visual (geometric) perspectives. This provides a comprehensive overview of any vector scenario, whether you're examining a static scenario or dynamic motion in physics.