Find the angle \( \theta \), in degrees, between \( \mathbf{u} \) and \( \mathbf{v} \). \( \begin{aligned} \mathbf{u} & =[7,0,6] \\ \mathbf{v} & =[3,8,4] \\ \theta & =[\mathrm{Ex}: 1.234\end{aligned}{ }^{\circ} \)

To find the angle \( \theta \) between the vectors \( \mathbf{u} = [7,0,6] \) and \( \mathbf{v} = [3,8,4] \), we can use the formula: \[ \cos(\theta) = \frac{\mathbf{u} \cdot \mathbf{v}}{||\mathbf{u}|| ||\mathbf{v}||} \] First, we calculate the dot product \( \mathbf{u} \cdot \mathbf{v} \): \[ \mathbf{u} \cdot \mathbf{v} = 7 \cdot 3 + 0 \cdot 8 + 6 \cdot 4 = 21 + 0 + 24 = 45 \] Next, we find the magnitudes of \( \mathbf{u} \) and \( \mathbf{v} \): \[ ||\mathbf{u}|| = \sqrt{7^2 + 0^2 + 6^2} = \sqrt{49 + 0 + 36} = \sqrt{85} \] \[ ||\mathbf{v}|| = \sqrt{3^2 + 8^2 + 4^2} = \sqrt{9 + 64 + 16} = \sqrt{89} \] Now, substituting back into the cosine formula: \[ \cos(\theta) = \frac{45}{\sqrt{85} \cdot \sqrt{89}} \] Calculating \( \sqrt{85} \) and \( \sqrt{89} \): - \( \sqrt{85} \approx 9.22 \) - \( \sqrt{89} \approx 9.43 \) So, \[ ||\mathbf{u}|| ||\mathbf{v}|| \approx 9.22 \cdot 9.43 \approx 87.02 \] Then, \[ \cos(\theta) \approx \frac{45}{87.02} \approx 0.517 \] Finally, we calculate \( \theta \): \[ \theta \approx \cos^{-1}(0.517) \approx 58.39^\circ \] Therefore, the angle \( \theta \) between \( \mathbf{u} \) and \( \mathbf{v} \) is approximately: \[ \theta \approx 58.39^\circ \]