Let S be the triangle with vertices at the origin and at a pair of vectors a,b $ \in \mathbf R^3 $ with a × b = ( 0, 3, 4) with unit normal vector n pointing in the direction a × b and let f=(1,2,1). What is the value of the surface integral $ \iint\limits_{S}\mathbf{f\cdot \mathbf{n}}dS $ ?
what I know so far:
The equation is the flux of f through S in the orientation n . I know the definition of the flux but don't know how to use it to compute this as its with a triangle i.e limits.
Any help would be appreciated.
Hint:
$\hat{n} = ( 0, \frac{3}{5}, \frac{4}{5})$ as the unit normal is in direction of $\vec{a} \times \vec{b}$.
$f = (1, 2, 1)$
Instead of a double integral and knowing its bounds, in this case you will use the area of the triangle that is known from the magnitude of the cross product of two vectors of the triangle $(|\vec{a} \times \vec{b}|$).