The Reason for the dot product in work done definition

It was defined the work as

W                =                Fext * d

You can see here that, there is no need of vector products here, because the displacement by the net external force will always be in the direction of the Net external force.  Then what is the problem?

The problem arises when the body is already in motion or already acted by an external force.

First we will take the case of most usual one i.e. pushing an object on Earth where the object is initially at rest.  

When you try to move an object in earth by pushing at an angle θ the object will move only in the horizontal direction.

Don’t think that Newton’s law is not valid here.

You may think that the net external force is not in the direction of the motion. But it is really in the direction of the motion.

What happens here is that the perpendicular component of the applied force would be canceled out by the Normal force from the ground.

Normal force is just the electrostatic repulsion forces between various objects which prevent one object move into another.

So, the net external force will only be in horizontal direction.

The problem of direction is not essential, if you again consider the work in terms of the net external force. But you cannot always consider the Normal force separately. 

We simply know that nothing is going into the Earth. So, all perpendicular components of the force should be canceled out.

It shows that the net external force is not important, but the component of the force along the direction of the motion is important.

Then,

W = Force along the direction of the motion * displacement

Now, we will go to the second case where the object itself initially moving at some velocity.   

Take an example of a projectile motion. 

When the object is thrown with an initial velocity, due to the net gravitational force acting downwards the projectile will take a parabolic path.

Though the projectile takes a curved path, you will consider only the displacement along the direction (vertical direction) of the net external force when you need to calculate the amount of work done. 

The effects in horizontal direction is not considered since there is no external force in that direction. 

It is used only the amount of work done in the direction of the force and that will only change the energy of the particle. Energy in other components of direction wouldn’t be affected.

Then,

W = Force* displacement along the direction of the force

For the sake of simplicity we just redefined the work as follows,

W = F*d*cos θ

Now this definition is suitable for all the cases. The angle  “θ” is defined from the directions of the force vector and the displacement vector by joining the tails of the both vectors.

Only to find the value of “cos θ”, we need the directions of Force vector and the displacement vector. And that is why, the work done is simply written using the vector notation as,

W = F⃑    . d⃑   = F*d*cos θ

Note: "F*d*cos θ" is only translated as  the dot product - F⃑    . d⃑   

and not F⃑    . d⃑   is translated as "F*d*cosθ". 

*There is a difference.

where “F” and “d” are respectively the magnitudes of the force and displacement vector. And the work done is just a magnitude. It can have either positive or negative values depending on the directions of force and displacement vector.

A simple analysis about why this formula is perfect as follows,

If an object is moved perpendicular to the direction of the force then the work done by that force on that direction is zero. The reasoning is simple,

“If I moved you, you will move only in the direction of my force or its component direction”. But you will never move in a direction perpendicular to my force. The perpendicular motion is independent from the force applied. The force will not be anyway responsible to the motion of the object in perpendicular direction.  

Don’t consider the example of doing work by placing a stone at your head and standing for a long time. With respect to Physics, it is not a work, because all the physics things defined for point particles. And you are not a point particle.