Question

find the angle between the vectors \\(\langle 7, -2 \
angle\\) and \\(\langle -5, -3 \
angle\\). carry your intermediate computations to at least 4 decimal places. round your final answer to the nearest degree.

Explanation:

Identify the given vectors and the formula

We are given two vectors:

$$\mathbf{u} = \langle 7, -2 angle$$

$$\mathbf{v} = \langle -5, -3 angle$$

To find the angle \(\theta\) between them, we use the dot product formula:

$$\cos(\theta) = \frac{\mathbf{u} \cdot \mathbf{v}}{\|\mathbf{u}\| \|\mathbf{v}\|}$$

Calculate the dot product

The dot product \(\mathbf{u} \cdot \mathbf{v}\) is calculated as:

$$\mathbf{u} \cdot \mathbf{v} = (7)(-5) + (-2)(-3)$$

$$\mathbf{u} \cdot \mathbf{v} = -35 + 6 = -29$$

Calculate the magnitudes of both vectors

Using the Vector Magnitude formula:

$$\|\mathbf{u}\| = \sqrt{7^2 + (-2)^2} = \sqrt{49 + 4} = \sqrt{53} \approx 7.2801$$

$$\|\mathbf{v}\| = \sqrt{(-5)^2 + (-3)^2} = \sqrt{25 + 9} = \sqrt{34} \approx 5.8310$$

Compute the cosine of the angle

Substitute the values into the cosine formula:

$$\cos(\theta) = \frac{-29}{\sqrt{53} \cdot \sqrt{34}} = \frac{-29}{\sqrt{1802}}$$

$$\sqrt{1802} \approx 42.44997$$

$$\cos(\theta) \approx \frac{-29}{42.44997} \approx -0.683157$$

Calculate the angle in degrees

Find the inverse cosine of the value:

$$\theta = \arccos(-0.683157)$$

$$\theta \approx 133.085^\circ$$

Rounding to the nearest degree gives:

$$\theta \approx 133^\circ$$

Answer:

Find the angle between the vectors \(\langle 7, -2
angle\) and \(\langle -5, -3
angle\). Carry your intermediate computations to at least 4 decimal places. Round your final answer to the nearest degree.

<blank>133</blank>\(^\circ\)