Area of a parallelogram using determinants

In summary, to find the area of the parallelogram formed by vectors v and u, you need to create two new vectors rv and ru by finding the difference between the origin and v, and the origin and u. Then, take the cross product of these two new vectors to get the magnitude of the area of the parallelogram. Remember to also divide the result by 2, as the area of a triangle formed by the two vectors is half the area of the parallelogram. It is important not to ignore any components given in the question, as they are necessary for finding the correct answer.
  • #1
cse63146
452
0

Homework Statement



let v = (1,0,1) and u = (0,2,1)

Find the area of the parallelogram {sv + tu : 0 <= s, t <=1)

Homework Equations





The Attempt at a Solution



I know the area of a parallelogram is the determinant of a 2x2 matrix, but they gave v and u in R^3. Would I just ignore the z component in this case?
 
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  • #2
technically there is a 3rd vector which could be r = (0,0,0)

and NEVER ignore any component given in a question like this :P

So, let the origin = r therefore find the vectors rv and ru
Then, find the magnitude of the cross product of the two vectors, rv and ru
i.e. |rv x ru|
Your answer should be the Area of the Parallelogram. The Area of a Triangle formed in vectors is HALF the Parallelogram.

I hope I've been helpful.

missbooty87
 
  • #3
I could be wrong but isn't (0,0,0)(1,0,1) = 0?
 
  • #4
cse63146 said:
I could be wrong but isn't (0,0,0)(1,0,1) = 0?

yes... But how is that relevant to your question... i said find the two new vectors and then cross multiply the two new vectors he he... not multiply or cross-multiply the individual vectors he he...

And if I wasn't clear let me rephrase.
--------------------
1st step:

find the two new vectors

the first vector is from R to V (i.e. From the Origin to the vector v)
the second vector is from R to U
--------------------
2nd step:

Cross multiply the RV and RU (i.e. |RV x RU|)
--------------------
3rd step:

Claim that you have the answer
--------------------

If you don't understand let me know.
 

1. How do you find the area of a parallelogram using determinants?

To find the area of a parallelogram using determinants, you can use the formula A = |ad - bc|, where a and b represent the coordinates of two adjacent sides of the parallelogram and c and d represent the coordinates of the other two adjacent sides.

2. What is the significance of using determinants to find the area of a parallelogram?

Using determinants to find the area of a parallelogram allows us to calculate the area using the coordinates of the parallelogram's sides instead of having to measure the sides directly. This method is also applicable to finding the area of other shapes in higher dimensions.

3. Can you use determinants to find the area of any parallelogram?

Yes, determinants can be used to find the area of any parallelogram, regardless of its size or orientation. This is because the formula for finding the area using determinants is based on the properties of parallelograms.

4. Is there a limit to the number of dimensions in which determinants can be used to find the area of a parallelogram?

No, determinants can be used to find the area of a parallelogram in any number of dimensions. However, the formula for calculating the area will change depending on the number of dimensions.

5. How does finding the area of a parallelogram using determinants differ from other methods?

Unlike other methods of finding the area of a parallelogram, using determinants does not require directly measuring the sides of the parallelogram. Instead, it uses the coordinates of the sides to calculate the area, which can be more efficient and accurate.

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