How to specify the direction of an area vector?

In summary: 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.
  • #1
Hawkingo
56
2
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?
 
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  • #2
Hawkingo said:
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}##.
 
  • #3
fresh_42 said:
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.
 
  • #4
Hawkingo said:
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.
 
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Related to How to specify the direction of an area vector?

1. How do I determine the direction of an area vector?

The direction of an area vector is determined by the right-hand rule. This means that if you curl your fingers in the direction of the edges of the area, your thumb will point in the direction of the area vector.

2. Can the direction of an area vector be negative?

No, the direction of an area vector cannot be negative. It is a vector quantity and therefore has both magnitude and direction. The direction can be positive or negative, but not the vector itself.

3. Does the direction of an area vector matter?

Yes, the direction of an area vector is important in determining the direction of a magnetic field or the direction of torque on an object. It also affects the sign of the flux through a surface.

4. How is the direction of an area vector related to the normal vector?

The direction of an area vector is always perpendicular to the surface it is associated with, and therefore it is always parallel to the surface's normal vector.

5. Can the direction of an area vector change?

Yes, the direction of an area vector can change if the orientation of the surface changes. For example, if a triangle is flipped over, the direction of its area vector will also flip. However, the magnitude of the area vector will remain the same.

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