Area of the parallelogram when diagonal vectors are given.

Member
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

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

caffeinemachine

Well-known member
MHB Math Scholar
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

$$\displaystyle \alpha = 2i+6j-k$$ and $$\displaystyle \beta= 6i-8j+6k$$
Hint: If the diagonals of a parallelogram are known then you can find the sides. Figure out how.

earboth

Active member
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

$$\displaystyle \alpha = 2i+6j-k$$ and $$\displaystyle \beta= 6i-8j+6k$$
Here is a slightly different way to calculate the area of a parallelogram:

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

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