Question

find the angle between the vectors \\(\langle 8, 7 \
angle\\) and \\(\langle 5, -2 \
angle\\). carry your intermediate computations to at least 4 decimal places. round your final answer to the nearest degree.

Explanation:

Calculate the dot product of the vectors

Using the Vector Dot Product knowledge point

$$ \mathbf{u} \cdot \mathbf{v} = (8)(5) + (7)(-2) = 40 - 14 = 26 $$

Calculate the magnitude of each vector

Using the Vector Magnitude knowledge point

$$
LATEXBLOCK0
$$

Calculate the cosine of the angle

Using the Angle Between Vectors knowledge point

$$ \cos(\theta) = \frac{\mathbf{u} \cdot \mathbf{v}}{\|\mathbf{u}\| \|\mathbf{v}\|} = \frac{26}{\sqrt{113} \cdot \sqrt{29}} = \frac{26}{\sqrt{3277}} \approx \frac{26}{57.2451} \approx 0.4542 $$

Find the angle in degrees

Using the Angle Between Vectors knowledge point

$$ \theta = \arccos(0.4542) \approx 62.9868^\circ \approx 63^\circ $$

Answer:

Find the angle between the vectors \(\langle 8, 7
angle\) and \(\langle 5, -2
angle\). Carry your intermediate computations to at least 4 decimal places. Round your final answer to the nearest degree.

<blank>63</blank>\(^\circ\)