How to specify the direction of an area vector?

[Hawkingo]

We all know that the area of a triangle having consecutive sides as $\vec { a }$ and $\vec { b }$ has the area $\frac { 1 } { 2 } | \vec { a } \times \vec { b } |$ but what is the direction of that area vector? I mean if we consider $\vec { a } \times \vec { b }$ that will be one direction and if we consider $\vec { b } \times \vec { a }$ then that will be the opposite direction but as we know an vector always has a particular direction so how to specify the direction of the area vector in this case?

 

[fresh_42]

We all know that the area of a triangle having consecutive sides as $\vec { a }$ and $\vec { b }$ has the area $\frac { 1 } { 2 } | \vec { a } \times \vec { b } |$ but what is the direction of that area vector? I mean if we consider $\vec { a } \times \vec { b }$ that will be one direction and if we consider $\vec { b } \times \vec { a }$ then that will be the opposite direction but as we know an vector always has a particular direction so how to specify the direction of the area vector in this case?

The direction is determined by the right hand rule: thumb $\vec{a}$, pointer $\vec{b}$, middle $\vec{a}\times \vec{b}$.

 

[Hawkingo]

The direction is determined by the right hand rule: thumb $\vec{a}$, pointer $\vec{b}$, middle $\vec{a}\times \vec{b}$.

I know but I want to ask that why consider $\vec{a}\times \vec{b}$ for the area of the triangle but not $\vec{b}\times \vec{a}$ ? The 2 cross products have different directions.

 

[fresh_42]

I know but I want to ask that why consider $\vec{a}\times \vec{b}$ for the area of the triangle but not $\vec{b}\times \vec{a}$ ? The 2 cross products have different directions.

Like with all things which can be oriented: make your choice! Why do we write debts as negative numbers and not the other way around? Why do we define $\int_a^b f(x)dx = F(b)-F(a)$ and not the other way around? It's only a convention, and in this case I find it suited compared with the formula behind: $(\vec{a}\times \vec{b})_1=+ \det\left(\begin{bmatrix}a_2& b_2\\a_3&b_3\end{bmatrix} \right)$, i.e. to start with a positive sign.