A vector quantity is one which has both magnitude and direction and also obeys laws of vector algebra

(a) now considering a length of wire bent into loop it has a magnitude that is its length but no specific fixed direction the length at each instant can have a unique direction but no unique direction of the whole loop so it cannot be represented by a vector as shown in figure

(b) Now an Plane Area can be described as a vector since it has a magnitude that’s the area measured in m² and the whole plane area can be described by a vector whose direction shows the direction of normal to area and for a plane area normal is unique so direction of the whole plane area is unique as shown in figure

Now in general a plane area with surface area S can be described in vector form as

Where denotes the area vector S denote the magnitude of surface area and is a unit vector in the direction of normal or we can say perpendicular to the plane area

(c) Now a sphere have volume and surface area now volume also does not have a definite direction and has only a magnitude measured in m³ at each instant volume of a body or sphere will have different direction so volume of sphere cannot be associated with sphere because the whole volume cannot be represented by a unique direction now if we take area into account the normal at any every point of sphere will have a different direction directed outwards normally there is no unique direction to specify the whole area of sphere as shown in figure

So, the sphere cannot be associated with a vector because it there is no unique direction which could specify whole sphere.