How to specify the direction of an area vector?

• B
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?

Mentor
2022 Award
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.

Mentor
2022 Award
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.

YYtian and Hawkingo