The task in the image asks to compute the directional angles (Richtungswinkel) $\alpha$, $\beta$, and $\gamma$ of the given vector:

$\vec{a} = \begin{pmatrix} -1 \\ 3 \\ 2 \end{pmatrix}$

The directional angles are the angles between the vector and the coordinate axes. These can be calculated using the formula:

$\cos(\theta_i) = \frac{a_i}{|\vec{a}|}$

where $\theta_i$ is the angle with the corresponding axis, $a_i$ is the $i$-th component of the vector, and $|\vec{a}|$ is the magnitude (length) of the vector $\vec{a}$.

Step 1: Calculate the magnitude of $\vec{a}$

$|\vec{a}| = \sqrt{(-1)^2 + 3^2 + 2^2} = \sqrt{1 + 9 + 4} = \sqrt{14}$

Step 2: Compute the cosines of the angles

For the x-axis ($\alpha$):

$\cos(\alpha) = \frac{-1}{\sqrt{14}} \quad \Rightarrow \quad \alpha = \cos^{-1}\left(\frac{-1}{\sqrt{14}}\right)$

For the y-axis ($\beta$):

$\cos(\beta) = \frac{3}{\sqrt{14}} \quad \Rightarrow \quad \beta = \cos^{-1}\left(\frac{3}{\sqrt{14}}\right)$

For the z-axis ($\gamma$):

$\cos(\gamma) = \frac{2}{\sqrt{14}} \quad \Rightarrow \quad \gamma = \cos^{-1}\left(\frac{2}{\sqrt{14}}\right)$

Step 3: Calculate the angles

Let's compute these values:

1. $\alpha = \cos^{-1}\left(\frac{-1}{\sqrt{14}}\right) \approx \cos^{-1}(-0.2673)$
2. $\beta = \cos^{-1}\left(\frac{3}{\sqrt{14}}\right) \approx \cos^{-1}(0.8018)$
3. $\gamma = \cos^{-1}\left(\frac{2}{\sqrt{14}}\right) \approx \cos^{-1}(0.5345)$

Now, the approximate results in degrees:

• $\alpha \approx 105.96^\circ$
• $\beta \approx 36.87^\circ$
• $\gamma \approx 57.12^\circ$

These are the directional angles $\alpha$, $\beta$, and $\gamma$.

Would you like further details or have any questions about this process? Here are five related questions to deepen your understanding:

1. What is the significance of directional angles in 3D vector analysis?
2. How is the magnitude of a vector in 3D space computed?
3. What are the key differences between directional cosines and directional angles?
4. How would the directional angles change if the vector components were negative or zero?
5. Can we calculate directional angles for a zero vector?

Tip: When calculating directional angles, ensure the vector is not a zero vector, as this would lead to undefined angles.