Question

Vector area is a vector quantity associated with each plane figure whose magnitude is
a) Any quantity and direction parallel to the plane
b) Any quantity and direction perpendicular to the plane
c) Equal to the area and direction parallel to the plane
d) Equal to the area and direction perpendicular to the plane

Answer 1

Hint: Area vector is a vector quantity related to the plane or curved surface. If the area of the surface is A and the normal unit vector to the plane or surface is $\hat{n}$ then the area vector is given as $A\hat{n}$. Now, get the correct option related to the area vector with the help of above definition.

Complete step-by-step answer:
As the word “vector” suggests that it should be any vector quantity so let us define it using the words only. We know a vector is something that has a magnitude and direction as well, and as we define the quantity area vector for a plane or curved surface, which is related to its area. So, area vector has magnitude equal to the area of the plane or solid curved surface on which it's drawn and it has direction perpendicular to the surface of the plane and perpendicular to the tangent for the curved surface at the point where the area vector is drawn. So, we can draw figures according to the definition as

So, area vector consists of the normal vector and the area of the plane and always remember that the vector should be perpendicular to the tangent for the curved surface at the point where the area vector is drawn. Hence, we get area vector as
$\text{Area vector}=A.\hat{n}$
Hence, vector area is a vector quantity associated with each plane figure whose magnitude is equal to the area of plane and direction perpendicular to the plane.
So, option (d) is correct.

Note: Normal vector $\hat{n}$ involved with area vector is a unit vector perpendicular to the plane i.e. it gives only direction, not magnitude. So, don’t confuse the $\hat{n}$ for the question related to area vectors. Area vectors are used with the questions related to electric flux, heat equations, magnetic fields etc. In physics and in 3-D geometry finite.
Planar surface in mathematics as well. Most area vectors have application in mechanics, electric field and flux in physics only.