https://r2.erweima.ai/mathbot/image/1727051709359wg5vqy82.jpg

Let's analyze the problem.

The image contains the following three vectors:

\[
\vec{v_1} = \begin{pmatrix} -1 \\ -1 \\ -4 \end{pmatrix}, \quad \vec{v_2} = \begin{pmatrix} 1 \\ 4 \\ -1 \end{pmatrix}, \quad \vec{v_3} = \begin{pmatrix} -1 \\ -3 \\ 2 \end{pmatrix}
\]

The tasks are:

1. **Finding the area of the parallelogram** generated by \(\vec{v_1}\) and \(\vec{v_2}\).
2. **Finding the volume of the parallelepiped** generated by \(\vec{v_1}, \vec{v_2},\) and \(\vec{v_3}\).

### Step 1: Area of the Parallelogram
The area of a parallelogram formed by two vectors \(\vec{v_1}\) and \(\vec{v_2}\) is the magnitude of their cross product:
\[
\text{Area} = |\vec{v_1} \times \vec{v_2}|
\]

### Step 2: Volume of the Parallelepiped
The volume of a parallelepiped formed by three vectors \(\vec{v_1}, \vec{v_2}, \vec{v_3}\) is the absolute value of the scalar triple product:
\[
\text{Volume} = |\vec{v_1} \cdot (\vec{v_2} \times \vec{v_3})|
\]

I'll calculate these next.The results are as follows:

1. The area of the parallelogram generated by \(\vec{v_1}\) and \(\vec{v_2}\) is approximately **17.97**.
2. The volume of the parallelepiped generated by \(\vec{v_1}, \vec{v_2},\) and \(\vec{v_3}\) is **8**.

Would you like further details on how these were calculated or have any additional questions?

Here are some related questions to consider:

1. How is the cross product of two vectors geometrically interpreted?
2. What is the relationship between the dot product and vector projections?
3. How would the result change if we used different vectors for the parallelogram?
4. How can you generalize these concepts to higher dimensions?
5. What are the conditions for vectors to form a parallelepiped with zero volume?

**Tip:** The scalar triple product also tells you if three vectors are coplanar—if the result is zero, the vectors lie in the same plane.

Find the area of the parallelogram generated by vectors v₁ and v₂ and the volume of the parallelepiped generated by vectors v₁, v₂, and v₃.

This problem involves calculating the area of the parallelogram formed by vectors v₁ and v₂, and the volume of the parallelepiped generated by vectors v₁, v₂, and v₃. The key concepts include the cross product and scalar triple product, both foundational in vector geometry.

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