Area of a triangle by vectors - getting different answers

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Homework Help Overview

The problem involves finding the area of a triangle defined by three points in three-dimensional space using vector methods. The original poster, Thomas, reports obtaining different area results depending on the combinations of vectors used in the calculations.

Discussion Character

  • Exploratory, Mathematical reasoning, Assumption checking

Approaches and Questions Raised

  • Thomas attempts to calculate the area using different vector combinations derived from the points A, B, and C, but encounters discrepancies in the results. Some participants question the accuracy of the vector calculations and suggest that errors may stem from computational methods.

Discussion Status

The discussion is ongoing, with participants providing feedback on the calculations. There is an exchange of different area results based on the vectors used, and some participants are offering corrections and insights into potential errors in the calculations. However, there is no explicit consensus on the correct area at this point.

Contextual Notes

Thomas mentions redoing his work and expresses confusion over the differing results, indicating a possible misunderstanding of the vector operations or calculations involved. There is also a suggestion that reliance on a calculator or computer algebra system may be contributing to the errors.

thomas49th
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Homework Statement


A = (7,6,8)
B = (2,3,-1)
C = (2, -7, 9)

Find the area of the triangle these 3 points make

I'm getting slightly different results for different combinations I use

if I take

CA = a - c = (5 , 13 , -1)
CB = b - c = (0, 10, -10)

Taking another combination
AB = b - a = (-5, -3, -9)
AC = c - a = (-5, -13, 1)

Taking the modulus of the cross product and halving yields similar, but different answers. I've check and redone my work

Why?

Thanks
Thomas
 
Last edited:
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You computed AB wrong, it should be (-5,-3,-9) (added changed 3 to -3 in second coordinate). Then you get the same area.
 
sorry that was a typo, not actually what I did :blushing: (i've edited it now)

When I did it the first combination I get 5 sqrt(246)
But when I did the second combination I got 5sqrt(233)

Do you concur?

Thanks
Thomas
 
No.

I get the cross product as: (-120,50,50) with both your combinations which gives us an area of 5sqrt(194)
 
The correct answer is [-120, 50, 50] for your cross products.

You are screwing up when you are doing your calculations (I'm assuming you are using a CAS to input the numbers and not doing it all on paper which is why you missed it.)

If you type in for CB = <0, +10, +10> and do the cross product of CA x CB you will end up with [140, -50, 50] which results in your 5[tex]\sqrt{246}[/tex].

However if you type in CB correctly = <0, +10, -10> you'll come up with the correct answer.

Do it all on paper, ignore the calculator or CAS and you'll see where you went wrong. Once done properly you'll see that AB x AC = CA x CB
 

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