Find the angle $\theta$ between $\mathbf{v}=-8 \mathbf{i}+6 \mathbf{j}$ and $\mathbf{w}=4 \mathbf{i}+6 \mathbf{j}$. If necessary, round to the nearest tenth of a degree.
The angle between $\mathbf{v}$ and $\mathbf{w}$ is
Answer
The final answer is \(\boxed{86.8}\) degrees.
Step 1 ：Define the vectors \(\mathbf{v}=-8 \mathbf{i}+6 \mathbf{j}\) and \(\mathbf{w}=4 \mathbf{i}+6 \mathbf{j}\).
Step 2 ：Calculate the dot product of \(\mathbf{v}\) and \(\mathbf{w}\), which is \(4\).
Step 3 ：Calculate the magnitudes of \(\mathbf{v}\) and \(\mathbf{w}\), which are \(10.0\) and \(7.211102550927978\) respectively.
Step 4 ：Calculate the angle in radians between \(\mathbf{v}\) and \(\mathbf{w}\) using the formula \(\cos^{-1}\left(\frac{\text{dot product}}{\text{magnitude}_v \times \text{magnitude}_w}\right)\), which gives \(1.5152978215491797\) radians.
Step 5 ：Convert the angle from radians to degrees and round to the nearest tenth, which gives \(86.8\) degrees.
Step 6 ：The final answer is \(\boxed{86.8}\) degrees.
