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To find the volume of a parallelepiped defined by three vectors **u**, **v**, and **w**, we can use the scalar triple product formula. Given:

\[
\mathbf{u} = [5, 1, 1], \quad \mathbf{v} = [4, 8, 1], \quad \mathbf{w} = [3, 2, 4]
\]

The volume \( V \) of the parallelepiped is the absolute value of the scalar triple product:

\[
V = |\mathbf{u} \cdot (\mathbf{v} \times \mathbf{w})|
\]

### Step 1: Calculate the cross product \( \mathbf{v} \times \mathbf{w} \)
Using the determinant method:

\[
\mathbf{v} \times \mathbf{w} = 
\begin{vmatrix}
\mathbf{i} & \mathbf{j} & \mathbf{k} \\
4 & 8 & 1 \\
3 & 2 & 4 \\
\end{vmatrix}
\]

Expanding the determinant:

\[
= \mathbf{i} (8 \cdot 4 - 1 \cdot 2) - \mathbf{j} (4 \cdot 4 - 1 \cdot 3) + \mathbf{k} (4 \cdot 2 - 8 \cdot 3)
\]

\[
= \mathbf{i} (32 - 2) - \mathbf{j} (16 - 3) + \mathbf{k} (8 - 24)
\]

\[
= \mathbf{i} (30) - \mathbf{j} (13) + \mathbf{k} (-16)
\]

So,

\[
\mathbf{v} \times \mathbf{w} = [30, -13, -16]
\]

### Step 2: Calculate the dot product \( \mathbf{u} \cdot (\mathbf{v} \times \mathbf{w}) \)
Now, we take the dot product of \( \mathbf{u} = [5, 1, 1] \) with \( \mathbf{v} \times \mathbf{w} = [30, -13, -16] \):

\[
\mathbf{u} \cdot (\mathbf{v} \times \mathbf{w}) = (5 \cdot 30) + (1 \cdot -13) + (1 \cdot -16)
\]

\[
= 150 - 13 - 16
\]

\[
= 121
\]

### Step 3: Take the absolute value
The volume is the absolute value of this result:

\[
V = |121| = 121
\]

### Final Answer:
The volume of the parallelepiped is \( \boxed{121} \).

Would you like more details or have any questions?

---

1. How is the scalar triple product defined geometrically?
2. What are the properties of the cross product that make it useful in volume calculations?
3. Why is the absolute value taken in the scalar triple product?
4. How would the volume change if one of the vectors were scaled?
5. What if two vectors were parallel—how would that affect the volume?

**Tip:** Remember, the scalar triple product is zero if the vectors are coplanar (lying in the same plane), which would mean the parallelepiped has no volume.

Regn ut volumet av parallelepipedet utspent av vektorene \( \mathbf{u} \), \( \mathbf{v} \) og \( \mathbf{w} \) hvor \( \mathbf{u} = [5, 1, 1] \), \( \mathbf{v} = [4, 8, 1] \), \( \mathbf{w} = [3, 2, 4] \).

Learn how to calculate the volume of a parallelepiped using vector algebra and the scalar triple product. Given vectors u = [5, 1, 1], v = [4, 8, 1], and w = [3, 2, 4], this problem illustrates the cross product and dot product methods. Ideal for students in Grades 10-12.

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