Problem 39 - Algebra and Trigonometry

Question

Find the direction angles of the given vector, rounded to the nearest degree. $$ \langle 2,3,-6\rangle $$

Step-by-Step Solution

Answer

The direction angles are approximately 73°, 65°, and 151°.

Step 1: Calculate the Magnitude of the Vector

To find the direction angles, we first need to calculate the magnitude of the vector \( \langle 2,3,-6 \rangle \). The magnitude \( ||\mathbf{v}|| \) is given by the formula:\[ ||\mathbf{v}|| = \sqrt{x^2 + y^2 + z^2} \] Substituting the vector components, we get:\[ ||\mathbf{v}|| = \sqrt{2^2 + 3^2 + (-6)^2} = \sqrt{4 + 9 + 36} = \sqrt{49} = 7 \]

Step 2: Determine the Direction Cosines

The direction cosines are the cosines of the angles that the vector makes with the coordinate axes. They are calculated as follows:\[ \cos \alpha = \frac{x}{||\mathbf{v}||}, \quad \cos \beta = \frac{y}{||\mathbf{v}||}, \quad \cos \gamma = \frac{z}{||\mathbf{v}||} \] Substituting the values, we have:\[ \cos \alpha = \frac{2}{7}, \quad \cos \beta = \frac{3}{7}, \quad \cos \gamma = \frac{-6}{7} \]

Step 3: Calculate the Direction Angles

The direction angles \( \alpha, \beta, \) and \( \gamma \) can be found by taking the inverse cosine of the direction cosines:\[ \alpha = \cos^{-1}\left(\frac{2}{7}\right), \quad \beta = \cos^{-1}\left(\frac{3}{7}\right), \quad \gamma = \cos^{-1}\left(\frac{-6}{7}\right) \] Use a calculator to find the angles in degrees. Rounding to the nearest degree results in:\[ \alpha \approx 73^\circ, \quad \beta \approx 65^\circ, \quad \gamma \approx 151^\circ \]

Key Concepts

Vector Magnitude

A key step in working with vectors is understanding what a vector magnitude represents. The magnitude of a vector \ \( \mathbf{v} = \langle x, y, z \rangle \ \) is essentially its "length." You can think of it as the distance from the origin to the point in space that the vector describes. Calculating it involves applying the Pythagorean theorem in three dimensions.

Here's how you calculate the magnitude of a vector \ \( \langle 2, 3, -6 \rangle \ \). The formula used is:

• \[ ||\mathbf{v}|| = \sqrt{x^2 + y^2 + z^2} \]

Inserting our values into this formula results in a series of calculations to get:

• \[ ||\mathbf{v}|| = \sqrt{2^2 + 3^2 + (-6)^2} \]
• \[ = \sqrt{4 + 9 + 36} \]
• \[ = \sqrt{49} = 7 \]

Understanding the magnitude is critical because it is used to find directional attributes, like where the vector is pointing in relation to coordinate axes.

Direction Cosines

Direction cosines are an essential concept when dealing with vectors, particularly in a three-dimensional space. They provide a way to quantify the orientation of the vector relative to the main axes of the coordinate system.

For a vector \( \mathbf{v} = \langle x, y, z \rangle \), the direction cosines \( \cos \alpha, \cos \beta, \) and \( \cos \gamma \) correspond to the angles the vector makes with the x, y, and z axes, respectively.

To find these, you divide each component of the vector by the magnitude of the vector:

• \[ \cos \alpha = \frac{x}{||\mathbf{v}||} \]
• \[ \cos \beta = \frac{y}{||\mathbf{v}||} \]
• \[ \cos \gamma = \frac{z}{||\mathbf{v}||} \]

For our vector \( \langle 2, 3, -6 \rangle \), the calculation goes as follows:

• \[ \cos \alpha = \frac{2}{7} \]
• \[ \cos \beta = \frac{3}{7} \]
• \[ \cos \gamma = \frac{-6}{7} \]

These cosines are important in identifying the specific angle measures which describe the vector's orientation in space.

Inverse Cosine

To find the actual direction angles from the direction cosines, one must use the inverse cosine function. This function is crucial because it allows you to transform the ratios provided by the direction cosines into measurable angles.

Through the inverse cosine, or arc cosine, you can find the angles \( \alpha, \beta, \) and \( \gamma \) that the vector makes with the coordinate axes. Here is how you apply it:

• \[ \alpha = \cos^{-1}\left(\frac{2}{7}\right) \]
• \[ \beta = \cos^{-1}\left(\frac{3}{7}\right) \]
• \[ \gamma = \cos^{-1}\left(\frac{-6}{7}\right) \]

Using a calculator to evaluate these inverse cosines gives approximate angle measures in degrees:

• \( \alpha \approx 73^\circ \)
• \( \beta \approx 65^\circ \)
• \( \gamma \approx 151^\circ \)

So, inverse cosine helps transition from a vector's component ratios to precise directional angles, allowing for complete spatial orientation understanding.