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The problem asks to find the acute angle between two lines represented by vector functions.

Let \( \vec{r_1}(t) = \langle -4, -5, -2 \rangle + t \langle 8, 7, 9 \rangle \)  
and  
\( \vec{r_2}(t) = \langle -4, -2, 5 \rangle + t \langle 4, 7, 0 \rangle \).

To solve this, we will:

1. Extract the direction vectors of the lines.
2. Use the dot product formula to find the angle between these vectors.
3. Convert the angle to degrees.

Let’s work through these steps.

### Step 1: Direction Vectors

The direction vector of the first line is \( \vec{d_1} = \langle 8, 7, 9 \rangle \),  
and the direction vector of the second line is \( \vec{d_2} = \langle 4, 7, 0 \rangle \).

### Step 2: Dot Product and Magnitudes

The dot product formula is:

\[
\vec{d_1} \cdot \vec{d_2} = | \vec{d_1} | | \vec{d_2} | \cos \theta
\]

where \( \theta \) is the angle between the vectors.

Let’s calculate the dot product and the magnitudes of the vectors first.import numpy as np

# Define the direction vectors
d1 = np.array([8, 7, 9])
d2 = np.array([4, 7, 0])

# Calculate the dot product
dot_product = np.dot(d1, d2)

# Calculate the magnitudes of the vectors
magnitude_d1 = np.linalg.norm(d1)
magnitude_d2 = np.linalg.norm(d2)

# Calculate the angle in radians using the dot product formula
cos_theta = dot_product / (magnitude_d1 * magnitude_d2)
theta_radians = np.arccos(cos_theta)

# Convert the angle to degrees
theta_degrees = np.degrees(theta_radians)

dot_product, magnitude_d1, magnitude_d2, theta_degreesHere are the calculations:

- Dot product of \( \vec{d_1} \) and \( \vec{d_2} \): \( 81 \)
- Magnitude of \( \vec{d_1} \): \( 13.93 \)
- Magnitude of \( \vec{d_2} \): \( 8.06 \)

Using these values, the acute angle between the lines is approximately \( 43.84^\circ \).

Would you like further details or have any questions?

Here are five related questions to explore further:

1. How do you calculate the dot product of two vectors geometrically?
2. What is the physical significance of the dot product between vectors?
3. Can two vectors with the same direction have a different magnitude?
4. How would the result change if the vectors were orthogonal?
5. Can we determine if the angle is obtuse using the dot product?

**Tip:** Always ensure the vectors are direction vectors (difference in coordinates) when finding the angle between two lines.

Find the acute angle between the lines: Let r1(t) = <-4, -5, -2> + t<8, 7, 9> and r2(t) = <-4, -2, 5> + t<4, 7, 0>.

This problem demonstrates how to calculate the acute angle between two lines using vector functions and the dot product formula. The solution involves finding the direction vectors, computing their dot product, and using trigonometric principles to find the angle in degrees. Suitable for college-level students studying vector algebra.

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