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To find the angle \(\theta\) between the vectors \(\vec{a}\) and \(\vec{b}\), we can use the formula:

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

### Step 1: Calculate \(\vec{a} \cdot \vec{b}\)
The dot product \(\vec{a} \cdot \vec{b}\) is given by:
\[
\vec{a} \cdot \vec{b} = (5)(-2) + (0)(5) + (-4)(-1)
\]
Calculating each term:
\[
\vec{a} \cdot \vec{b} = -10 + 0 + 4 = -6
\]

### Step 2: Calculate the magnitudes \( |\vec{a}| \) and \( |\vec{b}| \)

1. **Magnitude of \(\vec{a}\):**
   \[
   |\vec{a}| = \sqrt{5^2 + 0^2 + (-4)^2} = \sqrt{25 + 0 + 16} = \sqrt{41}
   \]

2. **Magnitude of \(\vec{b}\):**
   \[
   |\vec{b}| = \sqrt{(-2)^2 + 5^2 + (-1)^2} = \sqrt{4 + 25 + 1} = \sqrt{30}
   \]

### Step 3: Substitute into the formula for \(\cos \theta\)

\[
\cos \theta = \frac{-6}{\sqrt{41} \cdot \sqrt{30}}
\]

Calculating the denominator:

\[
\sqrt{41} \cdot \sqrt{30} = \sqrt{1230}
\]

Thus,

\[
\cos \theta = \frac{-6}{\sqrt{1230}}
\]

### Step 4: Find \(\theta\) by taking the arccosine

\[
\theta = \arccos \left( \frac{-6}{\sqrt{1230}} \right)
\]

After calculating this, we find that:

\[
\theta \approx 1.743 \text{ radians}
\]

---

Would you like further details on any step, or have any questions?

Here are some related questions for further practice:

1. How do you calculate the cross product of two vectors?
2. How would you find the angle between two vectors if given in terms of i, j, and k?
3. What is the geometric meaning of the dot product of two vectors?
4. How can you tell if two vectors are perpendicular using their dot product?
5. How do you calculate the projection of one vector onto another?

**Tip:** Remember that the dot product between two vectors is zero if they are perpendicular.

Let \( \vec{a} = (5, 0, -4) \) and \( \vec{b} = (-2, 5, -1) \). Find the angle between the vectors (in radians).

Learn how to find the angle between vectors \( \vec{a} = (5, 0, -4) \) and \( \vec{b} = (-2, 5, -1) \) using the dot product and magnitudes. This problem covers vector algebra concepts, such as calculating the dot product and using the cosine formula for angles between vectors. Suitable for students in Grades 11-12.

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