Question

given ( mathbf{v} = 2mathbf{i} - mathbf{j} ) and ( mathbf{w} = 7mathbf{i} + 5mathbf{j} ), find the angle between ( mathbf{v} ) and ( mathbf{w} ).
what is the angle between ( mathbf{v} ) and ( mathbf{w} )?
(type your answer in degrees. do not round until the final answer. then round to the nearest tenth as needed.)

Explanation:

Step1: Calculate dot product

$\mathbf{v} \cdot \mathbf{w} = (2)(7) + (-1)(5) = 14 - 5 = 9$

Step2: Find $||\mathbf{v}||$

$||\mathbf{v}|| = \sqrt{2^2 + (-1)^2} = \sqrt{4 + 1} = \sqrt{5}$

Step3: Find $||\mathbf{w}||$

$||\mathbf{w}|| = \sqrt{7^2 + 5^2} = \sqrt{49 + 25} = \sqrt{74}$

Step4: Solve for $\cos\theta$

$\cos\theta = \frac{\mathbf{v} \cdot \mathbf{w}}{||\mathbf{v}|| \ ||\mathbf{w}||} = \frac{9}{\sqrt{5}\sqrt{74}} = \frac{9}{\sqrt{370}} \approx 0.469$

Step5: Find $\theta$ in degrees

$\theta = \cos^{-1}(0.469) \approx 62.0^\circ$

Answer:

$62.0^\circ$