Area of triangle given 3 vectors pointing to vertices

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SUMMARY

The area of a triangle defined by three vectors A, B, and C originating from the same point is calculated using the formula: area = 1/2 |(B×C) + (C×A) + (A×B)|. The discussion highlights the importance of visualizing the vectors correctly and suggests referencing the vectors to the triangle's vertices to simplify calculations. Participants emphasized the need to subdivide the triangle into smaller triangles to compute the total area effectively. A correction was made regarding the vector cross products, clarifying that the correct terms should include A×B instead of A×C.

PREREQUISITES
  • Understanding of vector operations, specifically cross products
  • Familiarity with geometric concepts related to triangles
  • Basic knowledge of vector representation in a coordinate system
  • Ability to manipulate and visualize vectors in three-dimensional space
NEXT STEPS
  • Study vector cross product properties and applications in geometry
  • Learn how to derive the area of polygons using vector methods
  • Explore the concept of vector referencing in geometric problems
  • Investigate the implications of vector orientation on area calculations
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Students studying vector calculus, mathematicians interested in geometric applications, and educators teaching vector geometry concepts.

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Homework Statement



Three vectors A, B, C point from the origin O to the three corners of a triangle. Show that the area of the triangle is given by

area = \frac{1}{2}|(B\timesC) + (C\timesA) + (A\timesC)|

Homework Equations



area of triangle with sides a, b, c = \frac{1}{2}|a\timesc|


The Attempt at a Solution



I can't figure out how to go about it. I know I need to find the lengths between the vertices and use that equation, but I don't know how to make it look like the expected result. I'm totally lost.
 
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You won't get anywhere with this problem if you don't try something.
 
What is the area of the triangle between two vectors a, b? Can you subdivide the triangle formed by the endpoints of three vectors into smaller triangles in a useful way?
 
haruspex said:
What is the area of the triangle between two vectors a, b? Can you subdivide the triangle formed by the endpoints of three vectors into smaller triangles in a useful way?

So, the area of the triangle between two vectors (let's say A and B) is 0.5|Axb| right? I still don't see how I can use that to solve this. I can find the area of every triangle but the one I need.

EDIT: Alright, I was just being a dummy. I redrew my picture so that each of the vectors point away from each other (into three different quadrants) and I realized the if I added up all of the triangles (each of which being contained in the big triangle) I'd get the area of the big one. So the area is the sum of the smaller areas. Thanks!
 
Your area equation must have a typo: the vectors products CxA + AxC sum to zero!

To solve this problem reference the vectors to the vertices; for example (B-A) and (C-A). This gives you vectors with the lengths of two of the sides ... which you already know how to do.
 
UltrafastPED said:
Your area equation must have a typo: the vectors products CxA + AxC sum to zero!

To solve this problem reference the vectors to the vertices; for example (B-A) and (C-A). This gives you vectors with the lengths of two of the sides ... which you already know how to do.

Yep, you're right. One of those was supposed to be an A cross B. Thanks.
 

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