Find the angle between $\mathbf{u}=\langle-2,5\rangle$ and $\mathbf{v}=\langle-5,-2\rangle$ to the nearest tenth of a degree. The angle between $\mathbf{u}$ and $\mathbf{v}$ is (Round to the nearest tenth.)
Solution
Step 1 ：Given vectors \(\mathbf{u} = \langle -2, 5 \rangle\) and \(\mathbf{v} = \langle -5, -2 \rangle\).
Step 2 ：The dot product of \(\mathbf{u}\) and \(\mathbf{v}\) is calculated as \(\mathbf{u} \cdot \mathbf{v} = u_1v_1 + u_2v_2 = 0\).
Step 3 ：The magnitudes of \(\mathbf{u}\) and \(\mathbf{v}\) are calculated as \(||\mathbf{u}|| = \sqrt{u_1^2 + u_2^2} = 5.385164807134504\) and \(||\mathbf{v}|| = \sqrt{v_1^2 + v_2^2} = 5.385164807134504\) respectively.
Step 4 ：Substitute these values into the formula for \(\theta\) and calculate the angle in radians as \(\theta = \cos^{-1}\left(\frac{\mathbf{u} \cdot \mathbf{v}}{||\mathbf{u}|| ||\mathbf{v}||}\right) = 1.5707963267948966\) radians.
Step 5 ：Convert the angle from radians to degrees to get \(\theta = 90.0\) degrees.
Step 6 ：Final Answer: The angle between \(\mathbf{u} = \langle -2, 5 \rangle\) and \(\mathbf{v} = \langle -5, -2 \rangle\) is \(\boxed{90.0}\) degrees.
