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To solve for the acute angle between the lines \( l_1 \) and \( l_2 \), we can use the formula for the angle \( \theta \) between two lines with direction vectors \( \vec{a} \) and \( \vec{b} \):

\[
\cos \theta = \frac{|\vec{a} \cdot \vec{b}|}{|\vec{a}| |\vec{b}|}
\]

### Step 1: Identify Direction Vectors
The lines given are:
1. \( l_1: \vec{r} = \begin{pmatrix} 5 \\ -1 \\ 1 \end{pmatrix} + t \begin{pmatrix} 3 \\ 1 \\ -1 \end{pmatrix} \).
2. \( l_2: x = 2y = z - 3 \).

For \( l_1 \), the direction vector \( \vec{a} \) is:
\[
\vec{a} = \begin{pmatrix} 3 \\ 1 \\ -1 \end{pmatrix}
\]

For \( l_2 \), the equations \( x = 2y \) and \( x = z - 3 \) can be rewritten in parametric form. Let's assume \( y = t \), then:
\[
x = 2t \quad \text{and} \quad z = 2t + 3
\]

Thus, the direction vector for \( l_2 \) is:
\[
\vec{b} = \begin{pmatrix} 2 \\ 1 \\ 2 \end{pmatrix}
\]

### Step 2: Calculate the Dot Product \( \vec{a} \cdot \vec{b} \)
\[
\vec{a} \cdot \vec{b} = \begin{pmatrix} 3 \\ 1 \\ -1 \end{pmatrix} \cdot \begin{pmatrix} 2 \\ 1 \\ 2 \end{pmatrix} = (3 \cdot 2) + (1 \cdot 1) + (-1 \cdot 2) = 6 + 1 - 2 = 5
\]

### Step 3: Calculate the Magnitudes \( |\vec{a}| \) and \( |\vec{b}| \)
\[
|\vec{a}| = \sqrt{3^2 + 1^2 + (-1)^2} = \sqrt{9 + 1 + 1} = \sqrt{11}
\]
\[
|\vec{b}| = \sqrt{2^2 + 1^2 + 2^2} = \sqrt{4 + 1 + 4} = \sqrt{9} = 3
\]

### Step 4: Substitute into the Formula
\[
\cos \theta = \frac{|\vec{a} \cdot \vec{b}|}{|\vec{a}| |\vec{b}|} = \frac{|5|}{\sqrt{11} \cdot 3} = \frac{5}{3\sqrt{11}}
\]

### Step 5: Calculate \( \theta \) (Acute Angle)
To find \( \theta \), take the inverse cosine:
\[
\theta = \cos^{-1} \left( \frac{5}{3\sqrt{11}} \right)
\]

This expression gives the acute angle \( \theta \) between the two lines \( l_1 \) and \( l_2 \).

Would you like a detailed numerical evaluation of \( \theta \) or have any questions?

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### Related Questions
1. How do we convert parametric equations to direction vectors in 3D geometry?
2. What does it mean geometrically if the angle between two lines is zero?
3. How can we confirm if two lines in 3D space are parallel or perpendicular?
4. How do we find the point of intersection of two lines in vector form?
5. What are some applications of finding angles between lines in engineering?

#### Tip
In 3D geometry, when two lines are not parallel or perpendicular, the angle between them can help determine the orientation of planes formed by these lines.

Find the acute angle between the lines l1: r = (5, -1, 1) + t(3, 1, -1) and l2: x = 2y = z - 3.

Learn how to find the acute angle between two lines in 3D space with direction vectors. This problem involves using the dot product to calculate the angle between the lines and provides a step-by-step solution.

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