First, find the vectors BO and BF:

\(\mathbf{BO} = -8\mathbf{i} - 6\mathbf{j}\)

\(\mathbf{BF} = -6\mathbf{j} - 8\mathbf{i} + 7\mathbf{k} + 4\mathbf{i} + 2\mathbf{j} = -4\mathbf{i} - 4\mathbf{j} + 7\mathbf{k}\)

Calculate the dot product \(\mathbf{BF} \cdot \mathbf{BO}\):

\(\mathbf{BF} \cdot \mathbf{BO} = (-4)(-8) + (-4)(-6) = 32 + 24 = 56\)

Calculate the magnitudes:

\(|\mathbf{BF}| = \sqrt{(-4)^2 + (-4)^2 + 7^2} = \sqrt{16 + 16 + 49} = \sqrt{81} = 9\)

\(|\mathbf{BO}| = \sqrt{(-8)^2 + (-6)^2} = \sqrt{64 + 36} = \sqrt{100} = 10\)

Use the cosine formula:

\(\cos \theta = \frac{\mathbf{BF} \cdot \mathbf{BO}}{|\mathbf{BF}| \times |\mathbf{BO}|} = \frac{56}{9 \times 10} = \frac{56}{90} = \frac{28}{45}\)

Thus, the angle \(\theta = \cos^{-1}\left( \frac{28}{45} \right)\).