In my elementary physics textbook, it is mentioned that Surface Tension is a scalar quantity. Yet, during the derivation of the ascent formula, they have broken Surface Tension into rectangular components like vectors. The diagrams in my textbook are as follows:-
What is going on exactly? Is my textbook oversimplifying the things?
Surface tension $\gamma$ is not a force. Its proper name is the coefficient of surface tension. It is similar to the spring constant $k$ and pressure $p$. All three are scalars which define the constant of proportionality between two vectors :
F= k x
F= p A
T = $\gamma$ l.
Here x is a displacement, A is a plane area which has a direction defined by the outward normal to the plane, and l is a line segment which has a direction defined by the outward normal to the line in the local plane of the surface. These are all vectors.
What the diagrams are illustrating is not the scalar coefficient $\gamma$ but the vector force T caused by the surface tension phenomenon. The vector T can be resolved into components.
Surface tension is a scalar insofar as it is always oriented parallel to the surface. So the implied direction is along the surface. In your capillary tube example, the liquid-air surface tension is aligned with the curved surface of the liquid. Thus, where the surface of the liquid meets the wall of the capillary tube (at contact angle $\theta$) there is a force directed along the surface of the fluid at an angle $\theta$. The vertical component of this force is $T\cos\theta$, as shown in your figure.
What you have done is to resolve the forces acting at the surface of a liquid into components.
The surface molecules of a liquid have an average separation which is greater than the separation of those molecules in the bulk of the liquid.
If the surface is cut then to keep the molecules on either side of the cut from moving forces have to be applied on the molecules which are parallel to the surface and at right angles to the cut.
The component of the force which must be applied at right angles to the cut per unit length of cut is called the surface tension and the phenomena which results in the surface acting like a membrane in tension is called surface tension.
So when the surface tension is quoted as $70\, \rm mN\, m^{-1}$ this is a scalar quantity which enables one to relate the attractive force acting between surface molecules per unit length of surface.
This is also $70\, \rm mJ\, m^{-2}$ which is the surface energy density which can be thought of as the energy stored in the surface per unit area of surface.
You can see how these ideas can be written as vector equations but note that $\gamma$, the surface tension, is a scalar quantity in these equations and directions are defined using normal and tangential unit vectors.
