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 \(\gamma\) for the vector \(\mathbf{u} = -2\mathbf{i} + 7\mathbf{j} + 5\mathbf{k}\), we can use the following formulas: 1. Calculate the magnitude of the vector: \[ |\mathbf{u}| = \sqrt{(-2)^2 + 7^2 + 5^2} = \sqrt{4 + 49 + 25} = \sqrt{78} \approx 8.837 \] 2. Use the formulas for direction angles: \[ \alpha = \cos^{-1}\left(\frac{u_x}{|\mathbf{u}|}\right), \quad \beta = \cos^{-1}\left(\frac{u_y}{|\mathbf{u}|}\right), \quad \gamma = \cos^{-1}\left(\frac{u_z}{|\mathbf{u}|}\right) \] 3. Calculate the angles: \[ \alpha = \cos^{-1}\left(\frac{-2}{\sqrt{78}}\right) \approx 2.601 \text{ rad} \] \[ \beta = \cos^{-1}\left(\frac{7}{\sqrt{78}}\right) \approx 0.404 \text{ rad} \] \[ \gamma = \cos^{-1}\left(\frac{5}{\sqrt{78}}\right) \approx 0.850 \text{ rad} \] So rounding to three decimal places, we have: \(\alpha \approx 2.601 \text{ rad}\), \(\beta \approx 0.404 \text{ rad}\), \(\gamma \approx 0.850 \text{ rad}\).