Calculate the area of the triangle- Vector Calculus

In summary, when given two vectors and asked to find the area of the parallelogram they form, it is possible to let one vector be equal to the difference between two points and the other vector be equal to the difference between one of those points and a third point. This will give the same area as other choices of vectors, as long as the two given vectors are not parallel or anti-parallel. It is important to check for errors when the computed result does not agree with the published answer, as it is possible that the problem is over-specified or the published answer is incorrect.
  • #1
chwala
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Homework Statement
see attached.
Relevant Equations
Vector Calculus
##\dfrac {1}{2}####\left\| {v×w}\right\|##
This is the question,
1643162165186.png


Now to my question, supposing the vectors were not given, can we let ##V=\vec {RQ}## and ##W=\vec {RP}##? i tried using this and i was not getting the required area. Thanks...
 
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  • #2
That should give the same answer. Try computing it again.
 
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  • #3
You should get the same result.
 
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  • #4
Look up cross product, I think?
 
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  • #5
ok, let me do that again...talk later in the day...I will amend my latex too...laters...
 
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  • #6
valenumr said:
Look up cross product, I think?
I feel silly... The latex rendered slowly.
 
  • #7
I already made myself feel silly,but the problem is over specified. You are correct to assume any choice of vector. With only two given, the other is defined, and any choice should give the same answer.
 
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  • #8
Yeah yeah...Great guys same values ##155, 190, -29##...i will post working later. Bingo!
 
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  • #9
chwala said:
Yeah yeah...Great guys same values ##155, 190, -29##...i will post working later. Bingo!
A=##\dfrac{1}{2}\|(8,-5,10)×(7,-8,-15)\|=\dfrac{1}{2}(155,190,-29)=\dfrac{1}{2}\sqrt{{155^2+190^2+(-29)^2}}≈123.46##
 
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  • #10
chwala said:
A=##\dfrac{1}{2}\|(8,-5,10)×(7,-8,-15)\|=\dfrac{1}{2}(155,190,-29)=\dfrac{1}{2}\sqrt{{155^2+190^2+(-29)^2}}≈123.46##
Nit: The second expression reads as being 1/2 of the vector <155, 190, -29>. What you should have is 1/2 the norm or magnitude of that vector.
 
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  • #11
Mark44 said:
Nit: The second expression reads as being 1/2 of the vector <155, 190, -29>. What you should have is 1/2 the norm or magnitude of that vector.
@Mark44 ...you have keen eyes mate! :biggrin:...I will amend that later in the day...cheers
 
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  • #12
chwala said:
i tried using this and i was not getting the required area.

Charles Link said:
That should give the same answer. Try computing it again.
If I'm working a problem and my result doesn't agree with the published answer, the first thing I do is to make sure I'm working the same problem. If that checks out, I then check my work for errors. Although it's possible that the published answer is wrong, this doesn't happen all that often.
 
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  • #13
chwala said:
A=##\dfrac{1}{2}\|(8,-5,10)×(7,-8,-15)\|=\dfrac{1}{2}\|(155,190,-29)\|=\dfrac{1}{2}\sqrt{{155^2+190^2+(-29)^2}}≈123.46##

Amended post ##9##.
 
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FAQ: Calculate the area of the triangle- Vector Calculus

1. What is the formula for calculating the area of a triangle using vector calculus?

The formula for calculating the area of a triangle using vector calculus is A = 1/2 |(a x b)|, where a and b are two sides of the triangle and |a x b| represents the magnitude of the cross product between a and b.

2. How is vector calculus used to find the area of a triangle?

Vector calculus is used to find the area of a triangle by taking the cross product of two sides of the triangle and then finding the magnitude of the resulting vector. This magnitude represents the area of the parallelogram formed by the two sides, and since a triangle is half of a parallelogram, dividing by 2 gives the area of the triangle.

3. Can vector calculus be used to find the area of any type of triangle?

Yes, vector calculus can be used to find the area of any type of triangle, including equilateral, scalene, and right triangles. It only requires knowing the lengths of two sides of the triangle and the angle between them.

4. What are the advantages of using vector calculus to find the area of a triangle?

One advantage of using vector calculus to find the area of a triangle is that it can be used for any type of triangle, as opposed to other methods that may only work for specific types. Additionally, vector calculus is a powerful mathematical tool that can be applied to many other areas of science and engineering.

5. Are there any limitations to using vector calculus to find the area of a triangle?

One limitation of using vector calculus to find the area of a triangle is that it requires knowledge of vector operations and may be more complex than other methods for some individuals. Additionally, if the triangle is not defined by two sides and the angle between them, vector calculus may not be applicable.

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