Gracy, if you move the vector, it isn't the same vector. It might be parallel to the initial one and of the same magnitude, but where the force is applied is important. That picture you posted does state that moving the vector doesn't change it, but if you read on, it states that changing the position vector, at least in this case, does not change magnitude or direction. In a physical situation, the position is important.
Take this example, the earth goes around the sun. There's a force that pulls the earth towards the sun and if you were to draw a vector diagram it would be an arrow with the tail at the earth's centre of mass and the head pointing towards the sun. Now say you move this vector over to a point on the circumference of the earth. Then that force would be applying a torque on the earth which would make it spin much faster than it is right now and we would all be quite dizzy. But that isn't the case.
So you see, in a physical situation, the position of the vector does matter and you can't move it somewhere without changing the physics of the situation.
I see your confusion. The link is primarily teaching about vectors, but it does explicitly mention force and velocity as generic examples of vectors. There are some caveats that were not made clear in the article.
IF you are only interested in the motion of the center of mass THEN you can treat force as a free vector. Otherwise, force is a localized vector for exactly the reason that you showed. If you apply the same force (magnitude and direction) at different locations the center of mass will move the same, but the rotations and deformations will generally be different.
Another thing you'll should know is that torque is a vector quantity. It is the cross product of the distance vector and the force vector. If you don't know what a cross product is, look it up.
The important point here is the fact that you're using a distance VECTOR. So, even though, when you move your force vector as shown,your force vector is essentially the same, your distance vector has changed. Its magnitude is the same but its direction is opposite, hence you get the wrong direction for the torque and the two torqyes seem to cancel each other out.
Torques. The forces cancel each other out even in the first diagram where the force vector hasn't been moved. That's why the dipole doesn't have any translational motion. It only rotates. But in the first diagram,the torques add up, in the second they cancel each other out.
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