MHB Area of the parallelogram when diagonal vectors are given.

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To find the area of a parallelogram when the diagonals are given, the diagonals can be represented as vectors, specifically $$\alpha = 2i + 6j - k$$ and $$\beta = 6i - 8j + 6k$$. The area can be calculated using the formula $$A = \frac{1}{2} \cdot \| \vec{\alpha} \times \vec{\beta} \|$$. This approach allows for the determination of the area based on the cross product of the diagonal vectors. Understanding the relationship between the diagonals and the sides of the parallelogram is crucial for this calculation. The method effectively provides a means to compute the area without needing the side vectors directly.
Suvadip
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I can find the area of the parallelogram when two adjacent side vectors are given. But how to find the area of the parallelogram when diagonals of the parallelogram are given as

$$\alpha = 2i+6j-k$$ and $$\beta= 6i-8j+6k$$
 
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suvadip said:
I can find the area of the parallelogram when two adjacent side vectors are given. But how to find the area of the parallelogram when diagonals of the parallelogram are given as

$$\alpha = 2i+6j-k$$ and $$\beta= 6i-8j+6k$$
Hint: If the diagonals of a parallelogram are known then you can find the sides. Figure out how.
 
suvadip said:
I can find the area of the parallelogram when two adjacent side vectors are given. But how to find the area of the parallelogram when diagonals of the parallelogram are given as

$$\alpha = 2i+6j-k$$ and $$\beta= 6i-8j+6k$$

Here is a slightly different way to calculate the area of a parallelogram:

According to your question $$\alpha$$ and $$\beta$$ denote the diagonals of a parallelogram. Then the area is

$$A = \frac12 \cdot \| \vec {\alpha} \times \vec {\beta} \|$$
 
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