Area of Triangle: Solving 1/2 |( BxC + CxA + AxB )|

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SUMMARY

The area of a triangle formed by three vectors A, B, and C from the origin O can be expressed as 1/2 |(BxC + CxA + AxB)|. This formula utilizes the properties of the cross product to derive the area based on the vectors' orientations. The discussion emphasizes the importance of understanding the cross product and its geometric interpretation in calculating the area of a triangle in vector space.

PREREQUISITES
  • Understanding of vector operations, specifically cross products.
  • Familiarity with vector notation and components in three-dimensional space.
  • Knowledge of trigonometric functions, particularly sine and their geometric interpretations.
  • Basic principles of linear algebra related to vector spaces.
NEXT STEPS
  • Study the properties and applications of the cross product in vector calculus.
  • Learn about the geometric interpretation of vectors and their relationships in three-dimensional space.
  • Explore the derivation of area formulas for polygons using vector mathematics.
  • Investigate the implications of vector magnitudes and angles in calculating areas of geometric shapes.
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Students studying vector calculus, mathematicians interested in geometric interpretations, and educators teaching concepts of area calculation using vectors.

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Homework Statement



The three vectors A, B, C point from the origin O to the three corners of a triangle. Show that the area of a triangle = 1/2 |( BxC + CxA + AxB )|

Homework Equations



1/2 |a x b| = 1/2 |a||b|sin(alpha)


The Attempt at a Solution



My initial attempt at this problem was to take the cross product like (Bx, By, Bz) x (Cx, Cy, Cz) and seeing if anything canceled. Unfortunately nothing did (I also tried with a '2d' vector (Bx, By, 0) etc and nothing canceled either).


I believe I have to somehow use the equation I listed, but I'm not really sure what to do. It seems a little counterproductive to do the cross product three times. There aren't any simplifications that I can see. Anyone know where I should start?
 
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For a triangle of side |a| and |b| with included angle , what does \frac{1}{2} |a||b|sin\alpha give?
 
Hi jesuslovesu! :smile:

Hint: what is the area of triangle OAB? :smile:
 

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