Given $\mathbf{v}=-7 \mathbf{i}-6 \mathbf{j}$ and $\mathbf{w}=5 \mathbf{i}-\mathbf{j}$, find the angle between $\mathbf{v}$ and $\mathbf{w}$.

Step 1 ：Given vectors \(\mathbf{v}=-7 \mathbf{i}-6 \mathbf{j}\) and \(\mathbf{w}=5 \mathbf{i}-\mathbf{j}\), we are to find the angle between \(\mathbf{v}\) and \(\mathbf{w}\).

Step 2 ：The angle between two vectors can be found using the dot product formula: \[\mathbf{v} \cdot \mathbf{w} = ||\mathbf{v}|| ||\mathbf{w}|| \cos(\theta)\] where \(\theta\) is the angle between the vectors, \(||\mathbf{v}||\) and \(||\mathbf{w}||\) are the magnitudes of the vectors, and \(\mathbf{v} \cdot \mathbf{w}\) is the dot product of the vectors.

Step 3 ：We can rearrange this formula to solve for \(\theta\): \[\theta = \cos^{-1}\left(\frac{\mathbf{v} \cdot \mathbf{w}}{||\mathbf{v}|| ||\mathbf{w}||}\right)\]

Step 4 ：We can calculate the dot product \(\mathbf{v} \cdot \mathbf{w}\) as: \[\mathbf{v} \cdot \mathbf{w} = -7*5 + -6*(-1) = -35 + 6 = -29\]

Step 5 ：The magnitudes of the vectors are: \[||\mathbf{v}|| = \sqrt{(-7)^2 + (-6)^2} = \sqrt{49 + 36} = \sqrt{85}\] and \[||\mathbf{w}|| = \sqrt{5^2 + (-1)^2} = \sqrt{25 + 1} = \sqrt{26}\]

Step 6 ：So we can substitute these values into the formula for \(\theta\) to find the angle.

Step 7 ：Final Answer: The angle between vectors \(\mathbf{v}\) and \(\mathbf{w}\) is approximately \(\boxed{128.09}\) degrees.