Let $$ \vec{a}(t)=-4,4,-3\rangle t+\langle 1,2,-1\rangle $$ and $$ \vec{b}(u)=\langle 2,-3,-1\rangle u+\langle-3,8,-4\rangle $$. Find the acute angle (in degrees) between the lines:

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To find the acute angle between the two lines defined by the parametric equations:

$$
\vec{a}(t) = \langle -4, 4, -3 \rangle t + \langle 1, 2, -1 \rangle
$$
$$
\vec{b}(u) = \langle 2, -3, -1 \rangle u + \langle -3, 8, -4 \rangle
$$

we need to determine the angle between their direction vectors.

### Step 1: Identify the Direction Vectors
- **Direction vector of $$\vec{a}(t)$$:** $$\vec{v} = \langle -4, 4, -3 \rangle$$
- **Direction vector of $$\vec{b}(u)$$:** $$\vec{w} = \langle 2, -3, -1 \rangle$$

### Step 2: Compute the Dot Product
$$
\vec{v} \cdot \vec{w} = (-4)(2) + (4)(-3) + (-3)(-1) = -8 - 12 + 3 = -17
$$

Since we are interested in the acute angle, we take the absolute value:
$$
|\vec{v} \cdot \vec{w}| = 17
$$

### Step 3: Compute the Magnitudes of the Vectors
$$
|\vec{v}| = \sqrt{(-4)^2 + 4^2 + (-3)^2} = \sqrt{16 + 16 + 9} = \sqrt{41}
$$
$$
|\vec{w}| = \sqrt{2^2 + (-3)^2 + (-1)^2} = \sqrt{4 + 9 + 1} = \sqrt{14}
$$

### Step 4: Compute the Cosine of the Angle
$$
\cos(\theta) = \frac{|\vec{v} \cdot \vec{w}|}{|\vec{v}| |\vec{w}|} = \frac{17}{\sqrt{41} \cdot \sqrt{14}} = \frac{17}{\sqrt{574}} \approx \frac{17}{23.958} \approx 0.709
$$

### Step 5: Determine the Angle
Taking the arccosine:
$$
\theta = \arccos(0.709) \approx 45^\circ
$$

### Conclusion
The acute angle between the two lines is approximately **45 degrees**.

**Answer:** The acute angle between the lines is 45 degrees.
The acute angle between the two lines is 45 degrees.

Let and .
Find the acute angle (in degrees) between the lines:

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