Area of triangle formed by 2 vectors

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Homework Help Overview

The discussion revolves around calculating the area of a triangle formed by two vectors in three-dimensional space. The original poster presents vectors a and b and describes their relationship in forming triangle ABC at point P.

Discussion Character

  • Exploratory, Mathematical reasoning, Problem interpretation

Approaches and Questions Raised

  • Participants discuss the method of finding the area through the cross product of the vectors and the subsequent calculation of the area of the parallelogram. There is also an exploration of calculating the height from point P to line BC based on the area and the vectors.

Discussion Status

Some participants confirm the original poster's approach to calculating the area, while others provide additional methods for determining the height PM. Multiple interpretations of the problem are being explored, particularly regarding the use of trigonometric relationships.

Contextual Notes

There is mention of needing to find the angle between the vectors and the implications of the area calculation on further geometric properties, such as the height from point P to line BC. The discussion reflects uncertainty about the next steps in the calculation process.

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Homework Statement


I have a triangle ABC formed by vectors a = ( 2, 10, 25 ) and b = ( 5, 4, -2 ) attached at a point P.
And I have to calculate the area of the triangle ABC.


Homework Equations


I'm pretty certain that I need to find the area of the parellogram and then half it.
So I need to find the cross product, by calculating the determinant.


The Attempt at a Solution


Calculating the determinant I got
-120 i + 129 j - 42 k
so does this mean i square those and take the square root of them to find the area of the parellogram and then half it...
= 181.121
=> area of triangle = 90.56
 
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Sure, you're doing it right.
 
Thanks..I thought I was on the right lines, just needed reassuring :)

I got the area of the traingle to be 81root5 / 2

But now I need to calculate the length of the height PM (where PM perpendicular to BC and the point M belongs to the line BC)

I'm guessing as I know the area and 2 vectors I can take it from there...but I really have no idea...
 
Use |axb|=|a|*|b|*sin(t), where t is the smaller angle between the vectors a and b. Find t and use trig.
 

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