Can area of a triangle be 0.5(c x a) instead of 0.5(a x c)?

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Discussion Overview

The discussion revolves around the formula for calculating the area of a triangle using vector cross-products, specifically whether the area can be expressed as 0.5(c x a) instead of 0.5(a x c). The scope includes mathematical reasoning and technical explanation related to vector operations.

Discussion Character

  • Technical explanation, Mathematical reasoning

Main Points Raised

  • Some participants suggest that the area of a triangle can be calculated using the magnitude of the cross-product of two vectors, specifically stating that Area = (1/2)||A x C|| = (1/2)||C x A||.
  • Others clarify that the direction of the cross-product does not affect the area calculation since the magnitudes are the same, thus supporting the equivalence of the two expressions.
  • A participant questions the applicability of the formula in relation to the specific figure, noting that it may depend on the angle between the vectors, particularly if angle B is 90 degrees.

Areas of Agreement / Disagreement

There is some agreement on the use of the cross-product to find the area of a triangle, but there is also uncertainty regarding the conditions under which the formula applies, particularly related to the angles involved.

Contextual Notes

The discussion does not resolve the implications of different angles on the area calculation, nor does it clarify any assumptions regarding the vectors involved.

coconut62
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Please see the image attached.

Does that have anything to do with directions? The right-hand rule?
 

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Mathematics news on Phys.org
ca = ac, so yes.
 
Not in that figure, unless angle B is 90 degrees.
 
using the magnitude of the cross-product of two vectors to find the ar

If you mean

Area of triangle = (1/2)||A x C|| = (1/2)||C x A||

as in, using the magnitude of the cross-product of two vectors to find the area of the triangle between them, then yes.
 

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mark.watson said:
If you mean

Area of triangle = (1/2)||A x C|| = (1/2)||C x A||

as in, using the magnitude of the cross-product of two vectors to find the area of the triangle between them, then yes.

I didn't see the second part of your question. The direction of ||A x C|| or ||C x A|| doesn't matter because they have the same magnitude. That is why (1/2)||A x C|| = (1/2)||C x A|| is true.
 
Thank you very much.
 

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