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

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 \[ \boxed{ \begin{aligned} \alpha &= 1.801 \ \text{rad} \\ \beta &= 0.657 \ \text{rad} \\ \gamma &= 0.968 \ \text{rad} \end{aligned} } \]