Calculating the Area of a Parallelogram Using Diagonals: A Vector Approach

In summary, to find the area of a parallelogram with diagonals a = 3i + j − 2k and b = i − 3j + 4k, you cannot just take the cross product of the diagonals. However, by drawing two identical parallelograms and using the sum and difference of the diagonals, you can find the area using the formula |((A+B)/2) X ((A-B)/2)|, where (A+B)/2 is the base vector and (A-B)/2 is the side vector.
  • #1
Angello90
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Homework Statement


Find the area of the parallelogram with diagonals a = 3i + j − 2k and b = i − 3j + 4k


The attempt at a solution

I know that |x| X |y| will give the area, but will it hold for diagonals? Or do I have to find x and y vectors?
 
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  • #2
No, you can't just take the cross product of the diagonals. But if you draw two identical parallelograms side by side, you should be able to see that the sum of the two diagonals is twice the base vector. And putting one on top of the other, that the difference is twice the side vector.
 
  • #3
Ok so basically, |((A+B)/2) X ((A-B)/2)| = Area; where (A+B)/2 is a base and (A-B)/2 is a side?

Thanks a mil HallosofIvy!
 

FAQ: Calculating the Area of a Parallelogram Using Diagonals: A Vector Approach

What is the formula for finding the area of a parallelogram?

The formula for finding the area of a parallelogram is base x height. This can also be written as b x h or b * h, where b is the length of the base and h is the height of the parallelogram.

How do you measure the base and height of a parallelogram?

The base of a parallelogram is the length of one of its sides, and the height is the perpendicular distance from that side to the opposite side. You can measure the base and height using a ruler or measuring tape.

Can you use the same formula to find the area of any parallelogram?

Yes, the formula for finding the area of a parallelogram can be used for any parallelogram, regardless of its size or orientation. As long as you know the length of the base and the height of the parallelogram, you can use the formula to find its area.

How does the area of a parallelogram compare to the area of a rectangle?

A parallelogram and a rectangle may have the same base and height, but their area will be different. The area of a rectangle is length x width, while the area of a parallelogram is base x height. This means that a parallelogram with the same base and height as a rectangle will have a smaller area.

Can the area of a parallelogram be negative?

No, the area of a parallelogram cannot be negative. The area of a shape is always a positive value, representing the amount of space inside the shape. If you get a negative value when using the formula for the area of a parallelogram, it means you made a mistake in your calculations.

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