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The question asks us to:

1. Show that the area of a parallelogram with adjacent sides \(\vec{A}\) and \(\vec{B}\) is \(|\vec{A} \times \vec{B}|\).
2. Show that the area of a triangle with sides \(\vec{A}\) and \(\vec{B}\) is \(\frac{1}{2} |\vec{A} \times \vec{B}|\).

Let's break this down:

### 1. Area of a Parallelogram:
- The area of a parallelogram formed by two vectors \(\vec{A}\) and \(\vec{B}\) is given by the magnitude of their cross product.
- The formula for the area is:
  \[
  \text{Area of parallelogram} = |\vec{A} \times \vec{B}|
  \]
  where \(\vec{A} \times \vec{B}\) is the vector cross product, and the magnitude \( |\vec{A} \times \vec{B}| \) represents the area of the parallelogram formed by the two vectors.

### 2. Area of a Triangle:
- A triangle can be thought of as half of a parallelogram with the same base and height. Therefore, the area of a triangle formed by the same vectors \(\vec{A}\) and \(\vec{B}\) is half the area of the parallelogram.
- The area of the triangle is given by:
  \[
  \text{Area of triangle} = \frac{1}{2} |\vec{A} \times \vec{B}|
  \]

### Conclusion:
The area of a parallelogram with sides \(\vec{A}\) and \(\vec{B}\) is \(|\vec{A} \times \vec{B}|\), and the area of a triangle with the same sides is \(\frac{1}{2} |\vec{A} \times \vec{B}|\).

Would you like more details or further explanations?

Here are 5 related questions:
1. What is the physical interpretation of the cross product in vector geometry?
2. How can you calculate the cross product of two vectors given their components?
3. How does the cross product differ from the dot product in terms of geometry?
4. What happens to the area of the parallelogram if the vectors \(\vec{A}\) and \(\vec{B}\) are collinear?
5. Can you derive the formula for the area of a parallelogram using coordinate geometry?

**Tip:** Always remember that the magnitude of the cross product gives the area of the parallelogram, while the direction of the resulting vector is perpendicular to the plane of the parallelogram.

Show that the area of a parallelogram with sides A and B is |A × B|. Hence show that the area of a triangle with sides A and B is 1/2 |A × B|.

Learn how to calculate the area of a parallelogram and a triangle using the vector cross product. The problem covers the application of the formula |A × B| to find the area of a parallelogram and extends it to triangles.

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