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: $$\mathbf{v}\cdot \mathbf{w}=-7\ast 5+-6\ast (-1)=-35+6=-29$$

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 $\boxed{{\displaystyle 128.09}}$ degrees.