Find the direction angles of the vector. (Round your answers to three decimal places.) $$ \mathbf{u}=-2 \mathbf{i}+7 \mathbf{j}+5 \mathbf{k} $$ $$ \alpha=\square \mathrm{rad} $$ $$ \beta=\square \mathrm{rad} $$ $$ \gamma=\square \mathrm{rad} $$

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To find the direction angles $$ \alpha $$, $$ \beta $$, and $$ \gamma $$ of the vector $$ \mathbf{u} = -2 \mathbf{i} + 7 \mathbf{j} + 5 \mathbf{k} $$, follow these steps:

### Step 1: Compute the Magnitude of the Vector

The magnitude $$ |\mathbf{u}| $$ is given by:
$$
|\mathbf{u}| = \sqrt{(-2)^2 + 7^2 + 5^2} = \sqrt{4 + 49 + 25} = \sqrt{78} \approx 8.832
$$

### Step 2: Calculate the Direction Angles

The direction angles are the angles between the vector $$ \mathbf{u} $$ and the positive $$ x $$-, $$ y $$-, and $$ z $$-axes. They can be found using the following formulas:
$$
\alpha = \arccos\left(\frac{u_x}{|\mathbf{u}|}\right), \quad \beta = \arccos\left(\frac{u_y}{|\mathbf{u}|}\right), \quad \gamma = \arccos\left(\frac{u_z}{|\mathbf{u}|}\right)
$$

Substituting the components of $$ \mathbf{u} $$:

1. **For $$ \alpha $$:**
   $$
   \cos \alpha = \frac{-2}{8.832} \approx -0.226
   $$
   $$
   \alpha \approx \arccos(-0.226) \approx 1.801 \text{ radians}
   $$

2. **For $$ \beta $$:**
   $$
   \cos \beta = \frac{7}{8.832} \approx 0.793
   $$
   $$
   \beta \approx \arccos(0.793) \approx 0.657 \text{ radians}
   $$

3. **For $$ \gamma $$:**
   $$
   \cos \gamma = \frac{5}{8.832} \approx 0.566
   $$
   $$
   \gamma \approx \arccos(0.566) \approx 0.968 \text{ radians}
   $$

### Final Answer

$$
\boxed{
\begin{aligned}
\alpha &= 1.801 \ \text{rad} \
\beta  &= 0.657 \ \text{rad} \
\gamma &= 0.968 \ \text{rad}
\end{aligned}
}
$$
The direction angles are: - $$ \alpha = 1.801 $$ radians - $$ \beta = 0.657 $$ radians - $$ \gamma = 0.968 $$ radians

Find the direction angles of the vector. (Round your answers to three decimal places.)

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