## Finding Triangle Area using Cross Product

1. The problem statement, all variables and given/known data

Find the area of a triangle PQR, where P=(0,4,4), Q=(2,-6,-5), and R=(-3,-5,6)

3. The attempt at a solution

The vector PQ = (2, -10, -9)
The vector PR = (-3, -9, 2)

Using matrixes I set up something that looks like this...

I J K
2 -10 -9
-3 -9 2

Then using the matrix methods I get.

I(-20 - 81) - J(4 - 27) + K(-18 - 30)

I(-101) - J(-23) + K(48)

I take the square root of the squares and get.

109.4349122

What's the problem?
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 Blog Entries: 14 %was too lazy to do computations, so here's the matlab code (might help you) u = -3 -9 2 >> v v = 2 -10 -9 >> cross(v,u) ans = -101 23 -48 ans = 114.1665 >> ans/2 ans = 57.0833 why divided by 2? because axb = |a|.|b|.sin theta and area is 1/2 of that
 Recognitions: Gold Member Science Advisor Staff Emeritus The area of a parallelogram formed by two vectors, $\vec{u}$ and $\vec{v}$, is $|\vec{u}\times\vec{v}|$. Since a triangle is half a parallelogram, the the area of a triangle having sides $\vec{u}$ and $\vec{v}$ is half that.

## Finding Triangle Area using Cross Product

RootX, the square root of what you have square doesn't match up with the answer you have.

 -101 23 -48 ans = 114.1665 >> ans/2 ans = 57.0833 why divided by 2? because axb = |a|.|b|.sin theta and area is 1/2 of that
If you take the magnitude the negative numbers will become positive

$$\sqrt{(-101)^2 + (23)^2 + (-48)^2}= 114.167 ~ /2 = 57.0833$$