Can you associate vectors with $(a)$ the length of a wire bent into a loop,$(b)$ a plane area,$(c)$ a sphere? Explain.

(A) No; Yes; No.
$(a)$ The length of a wire bent into a loop is a scalar quantity. It has magnitude but no specific direction associated with it,so it cannot be represented as a vector.
$(b)$ $A$ plane area can be associated with a vector. The magnitude of the area vector is equal to the area of the plane,and its direction is defined as normal to the surface (either inward or outward).
$(c)$ $A$ sphere is a three-dimensional object defined by its volume,which is a scalar. Therefore,one cannot associate a vector with the volume of a sphere. However,an area vector can be associated with the surface area of a sphere at any specific point,directed radially outward.