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We use cross product of vector to find out area of a parallelogram and we end up with a vector.

As a learner, I know that area is nothing but the measure of the surface. So how can it have a particular direction?

Area does not seem to have directions like force, velocity, etc.

Qmechanic
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user100323
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6 Answers6

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Area is sometimes a vector because a number alone does not include all of the information we need about those things called areas. The additional information is "which way is the area facing" - direction, which is exactly the kind of information that vectors convey very well.

It may help to think about distance and displacement. In physics we distinguish between "distance" (not a vector) and "displacement" (the vector version) but the distinction is a not necessarily recognized outside of the field. There is no special term for "vector area" rather than "scalar area" as far as I know, but there probably could be.

fyodrpetrovich
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Area vector has a direction. It's direction is perpendicular to the surface.

We know Force= Pressure*Area

Now Force is a vector quantity , while pressure is a scalar quantity.

So it's basically a constant K (pressure ) getting multiplied by area vector to give force vector.

Note: it's not a dot product otherwise force would have been scalar.

Danny LeBeau
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  • Also there comes a notion of orientation of a surface, I think this answer could expand on that as it's an important notion in differential geometry and vector calculus – Triatticus Jun 25 '20 at 04:43
  • The example you picked may be a little more complex than you meant it to be. Pressure is three components of the Cauchy stress tensor. It still illustrates your point. – mmesser314 Jun 25 '20 at 14:25
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You may have been taught to skip a step. The cross product of two vectors does not equal the area of the parallelogram between them. It yields a vector whose magnitude is that area, and whose direction is defined to be normal to the surface. This magnitude is, of course, a scalar, as one would expect for an area.

As others have mentioned, the direction of this cross product is often useful for other reasons. For example, many equations end up taking the dot product between a flow or force and the normal to a surface. Using the cross product can make such equations much simple to evaluate.

Cort Ammon
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When a surface is embedded in a three dimensional space then the area can be represented by an arrow that is normal to the surface.

flippiefanus
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Area is an antisymmetric tensor. In 3D this has three independent components and behaves like a vector, except under inversion.

my2cts
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Area does have a direction, and it is perpendicular to the surface. However, if we think of a parallelogram in a 2-D space, then the resultant area would be a pseudovector since it would not lie in the same space.

Andrew Micheal Anderson
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