Area of triangle given 3 vectors pointing to vertices

Click For Summary

Homework Help Overview

The discussion revolves around finding the area of a triangle defined by three vectors A, B, and C originating from a common point. The original poster presents a formula for the area involving cross products of these vectors but expresses confusion about how to derive this result from known equations.

Discussion Character

  • Exploratory, Assumption checking, Problem interpretation

Approaches and Questions Raised

  • Participants explore the relationship between the area of triangles formed by pairs of vectors and question how to subdivide the triangle into smaller triangles to facilitate the calculation. There is also discussion about the validity of the area formula presented by the original poster.

Discussion Status

Some participants have offered guidance on referencing the vectors to the triangle's vertices and suggested methods to approach the problem. There is an acknowledgment of a potential typo in the area formula, prompting further clarification and exploration of the correct relationships between the vectors.

Contextual Notes

The original poster indicates a lack of clarity in applying the area formula and expresses uncertainty about how to manipulate the vectors to achieve the expected result. There is also a mention of redrawing the vectors for better visualization, which may influence the understanding of the problem.

majinsock
Messages
11
Reaction score
0

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.
 
Physics news on Phys.org
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.
 

Similar threads

  • · Replies 14 ·
Replies
14
Views
6K
  • · Replies 2 ·
Replies
2
Views
1K
  • · Replies 1 ·
Replies
1
Views
1K
Replies
15
Views
2K
  • · Replies 5 ·
Replies
5
Views
2K
Replies
3
Views
1K
  • · Replies 3 ·
Replies
3
Views
963
  • · Replies 3 ·
Replies
3
Views
3K
  • · Replies 2 ·
Replies
2
Views
2K
Replies
2
Views
2K