https://file.aiquickdraw.com/mathbot/file/17266202220680v8st1b8.jpg

Here are the visual representations of the problems:

1. **Triangle (Problem 5)**: The vectors \(\vec{a}\) (red) and \(\vec{b}\) (blue) form the two sides of the triangle. The area of the triangle is \(0.87\) square units.

2. **Parallelepiped (Problem 6)**: The three vectors \(\vec{A}\) (red), \(\vec{B}\) (green), and \(\vec{C}\) (blue) define the edges of the parallelepiped. The volume of this shape is \(20\) cubic units.

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### Follow-up questions:
1. How is the cross product used to find the area of a parallelogram or triangle?
2. Can the scalar triple product be used to find the volume of any shape?
3. How do vector magnitudes influence the shape of the parallelepiped?
4. What is the physical interpretation of the dot and cross products in real-world problems?
5. How would you extend this method to higher-dimensional vector spaces?

**Tip:** The cross product is only defined in three dimensions, but the concept of area and volume extends to higher dimensions with more general forms of the cross product.

Explique cómo resolver el área de un triángulo si tengo los siguientes vectores: a = i - 2j - k y b = -2i + 3j + k. Explique paso a paso la resolución del volumen de un paralelepípedo, cuyos vectores concurrentes son: A = i + 3j + 2k; B = 2i + j - k; C = i - 2j + k.

Learn how to calculate the area of a triangle using vector cross products and the volume of a parallelepiped using scalar triple products. Step-by-step explanations with examples for vector operations. Suitable for college students studying vector algebra and geometry.

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