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On the other hand, if the vectors labelled "HCL", "VCL", and "Total Lift" are only supposed to represent the horizontal, vertical, and total components of the wing's lift vector, then the diagrams become very confusing because the vector representing the aerodynamic sideforce generated by the airflow striking the side of the fuselage has been entirely omitted. In this case the vectors labelled "centrifugal force" should still appear as described in the above paragraph, but in the slipping and skidding cases, the vector labelled "HCL" will no longer be a mirror image of the vector labelled "centrifugal force". For a given bank angle, the wing's total lift vector, and therefore the horizontal and vertical components of the wing's lift vector, are all slightly smaller in a slipping turn than in a coordinated turn, because the aerodynamic sideforce generated by the airflow striking the side of the fuselage supports a small portion of the aircraft's weight.³ Similarly, in a skidding turn, the wing's total lift vector, and therefore the horizontal and vertical components of the wing's lift vector, are all slightly larger than in a coordinated turn, because the aerodynamic sideforce generated by the airflow striking the side of the fuselage contains an earthward component that must be opposed by the wing's lift vector. These differences should be apparent in the vectors labelled "VCL" and "Total Lift" as well as the vectors labelled "HCL", and the vector labelled "Total Lift" should be "square" to the wingspan in all three cases.

On the other hand, if the vectors labelled "HCL", "VCL", and "Total Lift" are only supposed to represent the horizontal, vertical, and total components of the wing's lift vector, then the diagrams become very confusing because the vector representing the aerodynamic sideforce generated by the airflow striking the side of the fuselage has been entirely omitted. In this case the vectors labelled "centrifugal force" should still appear as described in the above paragraph, but in the slipping and skidding cases, the vector labelled "HCL" will no longer be a mirror image of the vector labelled "centrifugal force". For a given bank angle, the wing's total lift vector, and therefore the horizontal and vertical components of the wing's lift vector, are all slightly smaller in a slipping turn than in a coordinated turn, because the aerodynamic sideforce generated by the airflow striking the side of the fuselage supports a small portion of the aircraft's weight.³ Similarly, in a skidding turn, the wing's total lift vector, and therefore the horizontal and vertical components of the wing's lift vector, are all slightly larger than in a coordinated turn, because the aerodynamic sideforce generated by the airflow striking the side of the fuselage contains an earthward component that must be opposed by the wing's lift vector. These differences should be apparent in the vectors labelled "VCL" and "Total Lift" as well as the vectors labelled "HCL", and the vector labelled "Total Lift" should be "square" to the wingspan in all three cases.

Rather, slips and skids are could be said to be characterized by an imbalance between the pseudoforce called "centrifugal force", and the horizontal component of the wing's lift force. (This isn't the most simple or most intuitive definition of a slip or skid, but it is a valid one.)

Since the pseudoforce called "centrifugal force" is really just a mirror image of the horizontal component of the net aerodynamic force generated by the aircraft, the above statement is just a really complicated way of saying that in a slip or skid, something other than the wing's lift vector is generating an aerodynamic force which has a horizontal component.

That "something" is the fuselage. In a slip or skid, the airflow is striking the side of the fuselage, which generates a real aerodynamic sideforce, oriented perpendicular to the wing's lift vector. Because the fundamental defining characteristic of a slip or skid is that the nose of the aircraft is not aligned with the actual direction of travel though the airmass, in the yaw axis.

On the other hand, if the vectors labelled "HCL", "VCL", and "Total Lift" are only supposed to represent the horizontal, vertical, and total components of the wing's lift vector, then the diagrams become very confusing because the vector representing the aerodynamic sideforce generated by the airflow striking the side of the fuselage has been entirely omitted. In this case the vectors labelled "centrifugal force" should still appear as described in the above paragraph, but in the slipping and skidding cases, the vector labelled "HCL" will no longer be a mirror image of the vector labelled "centrifugal force". For a given bank angle, the wing's total lift vector, and therefore the horizontal and vertical components of the wing's lift vector, are all slightly smaller in a slipping turn than in a coordinated turn, because the aerodynamic sideforce generated by the airflow striking the side of the fuselage supports a small portion of the aircraft's weight.³ Similarly, in a skidding turn, the wing's total lift vector, and therefore the horizontal and vertical components of the wing's lift vector, are all slightly larger than in a coordinated turn, because the aerodynamic sideforce generated by the airflow striking the side of the fuselage contains an earthward component that most be opposed by the wing's lift vector. These differences should be apparent in the vectors labelled "VCL" and "Total Lift" as well as the vectors labelled "HCL", and the vector labelled "Total Lift" should be "square" to the wingspan in all three cases.

Because when include the pseudoforce called "centrifugal force" in our vector diagrams, we are using an aircraft-based reference frame rather than an airmass-based reference frame or ground-based reference frame. Since the aircraft can't accelerate relative to itself, the net force in the aircraft-based reference frame will always be zero, whether the aircraft is turning or not. In the aircraft-based reference frame, the fact that the centrifugal force vector exists at all is actually evidence that the aircraft is turning. But for teaching purposes, using the aircraft-based reference frame (and therefore including the "centrifugal force" vector whenever the flight path is not linear) is arguably an inferior approach to using the airmass-based reference frame (and therefore not including any "centrifugal force" vector.)

Rather, slips and skids can be said to be characterized by an imbalance between the pseudoforce called "centrifugal force", and the horizontal component of the wing's lift force. (This isn't the simplest or most intuitive definition of a slip or skid, but it is a valid one.)

Since the pseudoforce called "centrifugal force" is actually just a mirror image of the horizontal component of the net aerodynamic force generated by the aircraft, the above statement is really just an overly complicated way of saying that in a slip or skid, something other than the wing is generating an aerodynamic force which has a horizontal component.

That "something" is the fuselage. In a slip or skid, the airflow is striking the side of the fuselage, which generates a real aerodynamic sideforce, oriented perpendicular to the wing's lift vector. Because the fundamental defining characteristic of a slip or skid is that the nose of the aircraft is not aligned with the actual direction of travel through the airmass, in the yaw axis.

On the other hand, if the vectors labelled "HCL", "VCL", and "Total Lift" are only supposed to represent the horizontal, vertical, and total components of the wing's lift vector, then the diagrams become very confusing because the vector representing the aerodynamic sideforce generated by the airflow striking the side of the fuselage has been entirely omitted. In this case the vectors labelled "centrifugal force" should still appear as described in the above paragraph, but in the slipping and skidding cases, the vector labelled "HCL" will no longer be a mirror image of the vector labelled "centrifugal force". For a given bank angle, the wing's total lift vector, and therefore the horizontal and vertical components of the wing's lift vector, are all slightly smaller in a slipping turn than in a coordinated turn, because the aerodynamic sideforce generated by the airflow striking the side of the fuselage supports a small portion of the aircraft's weight.³ Similarly, in a skidding turn, the wing's total lift vector, and therefore the horizontal and vertical components of the wing's lift vector, are all slightly larger than in a coordinated turn, because the aerodynamic sideforce generated by the airflow striking the side of the fuselage contains an earthward component that must be opposed by the wing's lift vector. These differences should be apparent in the vectors labelled "VCL" and "Total Lift" as well as the vectors labelled "HCL", and the vector labelled "Total Lift" should be "square" to the wingspan in all three cases.

Because when we include the pseudoforce called "centrifugal force" in our vector diagrams, we are using an aircraft-based reference frame rather than an airmass-based reference frame or ground-based reference frame. Since the aircraft can't accelerate relative to itself, the net force in the aircraft-based reference frame will always be zero, whether the aircraft is turning or not. In the aircraft-based reference frame, the fact that the centrifugal force vector exists at all is actually evidence that the aircraft is turning. But for teaching purposes, using the aircraft-based reference frame (and therefore including the "centrifugal force" vector whenever the flight path is not linear) is arguably an inferior approach to using the airmass-based reference frame (and therefore not including any "centrifugal force" vector.)

Those diagrams, versions of which have been reproduced in many different flight training "ground school" materials, including some published by the FAA, are extremely misleading, and contain errors and/or omissions, depending on exactly what each force vector is supposed to represent.

Rather, slips and skids are characterized by an imbalance between the pseudoforce called "centrifugal force", and the horizontal component of the wing's lift force.

If the vectors labelled "HCL" are supposed to represent the horizontal component of the net aerodynamic force generated by the aircraft, including the horizontal component of the aerodynamic sideforce generated by the airflow striking the side of the fuselage, then they are drawn incorrectly. They should be mirror images of the vectors labelled "centrifugal force".

On the other hand, if the vectors labelled "HCL" are only supposed to represent the horizontal component of the wing's lift vector, then they are drawn correctly, but the diagrams are very misleading because the vector representing the horizontal component of the aerodynamic sideforce generated by the airflow striking the side of the fuselage has been entirely omitted. Also, if the vectors labelled "HCL" are only supposed to represent the horizontal component of the wing's lift vector, then by logical extension the vectors labelled "VCL" must be intended to only represent the vertical component of the wing's lift vector, not the vertical component of the net aerodynamic force generated by the aircaft. In which case the vectors labelled "VCL" should not be all the same length, because in a slip, the aerodynamic sideforce from the air striking the side of the fuselage includes an upward vertical component which supports some of the aircraft weight and reduces the vertical force that the wing must generate, while in a skid, the aerodynamic sideforce from the air striking the side of the fuselage includes a downward vertical component which increase the vertical force that the wing must generate.

Obviously the diagrams would be greatly improved by changing to an airmass-based reference frame rather than an aircraft-based reference frame, so that the "centrifugal force" vector could be entirely discarded, and by also including the aerodynamic sideforce vector generated by the air striking the side of the fuselage. There's no need to break things into horizontal and vertical components, just show the wing's lift vector and the aerodynamic sideforce vector from the airflow striking the side of the fuselage. These two vectors are oriented perpendicular to each other. In a coordinated turn, there is no airflow striking the side of the fuselage, so the aerodynamic sideforce vector is zero, so the net aerodynamic force acts "straight up" in the aircraft's reference frame. In a slip or a skid, the aerodynamic sideforce vector is not zero, and so the net aerodynamic force does not act "straight up" in the aircraft's reference frame. End of story.

Because when include the pseudoforce called "centrifugal force" in our vector diagrams, we are using an aircraft-based reference frame rather than an airmass-based reference frame or ground-based reference frame. Since the aircraft can't accelerate relative to itself, the net force in the aircraft-based reference frame will always be zero, whether the aircraft is turning or not. In the aircraft-based reference frame, the fact that the centrifugal force vector exists at all is actually evidence that the aircraft is turning. You can see from the preceding parts of this answer that the author of this answer feels that using the aircraft-based reference frame (and therefore including the "centrifugal force" vector whenever the flight path is not linear) is an inferior approach to using the airmass-based reference frame (and therefore not including any "centrifugal force" vector.)

1. In this answer, when we talk about "horizontal", we mean as seen looking at the aircraft in a head-on view. We're not referring to fore-and-aft forces such as thrust and drag.

Those diagrams, versions of which have been reproduced in many different flight training "ground school" materials, including some published by the FAA, are extremely misleading, and contain errors and omissions.

Rather, slips and skids are could be said to be characterized by an imbalance between the pseudoforce called "centrifugal force", and the horizontal component of the wing's lift force. (This isn't the most simple or most intuitive definition of a slip or skid, but it is a valid one.)

If the vectors labelled "HCL" are supposed to represent the horizontal component of the net aerodynamic force generated by the aircraft, including the horizontal component of the aerodynamic sideforce generated by the airflow striking the side of the fuselage, then they are drawn incorrectly. For a given bank angle, the horizontal component of the net aerodynamic force generated by the aircraft is larger in a skidding turn than in a coordinated turn, and is smaller in a slipping turn than in a coordinated turn. If the vector labelled "total lift" is supposed to represent the total aerodynamic force generated by the aircraft, including the sideforce contribution from the airflow striking the side of the fuselage, then the coordinated turn is the only case where the "total lift" vector should be exactly "square" to the wingspan. In all cases the "centrifugal force" vector should be a mirror image of the vector representing the horizontal component of the net aerodynamic force generated by the aircraft.²

On the other hand, if the vectors labelled "HCL", "VCL", and "Total Lift" are only supposed to represent the horizontal, vertical, and total components of the wing's lift vector, then the diagrams become very confusing because the vector representing the aerodynamic sideforce generated by the airflow striking the side of the fuselage has been entirely omitted. In this case the vectors labelled "centrifugal force" should still appear as described in the above paragraph, but in the slipping and skidding cases, the vector labelled "HCL" will no longer be a mirror image of the vector labelled "centrifugal force". For a given bank angle, the wing's total lift vector, and therefore the horizontal and vertical components of the wing's lift vector, are all slightly smaller in a slipping turn than in a coordinated turn, because the aerodynamic sideforce generated by the airflow striking the side of the fuselage supports a small portion of the aircraft's weight.³ Similarly, in a skidding turn, the wing's total lift vector, and therefore the horizontal and vertical components of the wing's lift vector, are all slightly larger than in a coordinated turn, because the aerodynamic sideforce generated by the airflow striking the side of the fuselage contains an earthward component that most be opposed by the wing's lift vector. These differences should be apparent in the vectors labelled "VCL" and "Total Lift" as well as the vectors labelled "HCL", and the vector labelled "Total Lift" should be "square" to the wingspan in all three cases.

Obviously the diagrams would be greatly improved by changing to an airmass-based reference frame rather than an aircraft-based reference frame, so that the "centrifugal force" vector could be entirely discarded, and by also including by the aerodynamic sideforce vector generated by the air striking the side of the fuselage. There's no need to break things into horizontal and vertical components-- just show the wing's lift vector and the aerodynamic sideforce vector from the airflow striking the side of the fuselage. These two vectors are oriented perpendicular to each other. In a coordinated turn, there is no airflow striking the side of the fuselage, so the aerodynamic sideforce vector is zero, so the net aerodynamic force acts "straight up" in the aircraft's reference frame. In a slip or a skid, the aerodynamic sideforce vector is not zero, and so the net aerodynamic force does not act "straight up" in the aircraft's reference frame. End of story. (If desired, an "apparent load" vector could be added, which would always be the mirror image of the vector sum of the wing's lift vector and the aerodynamic sideforce vector from the airflow striking the side of the fuselage. Only in the case of the coordinated turn, where aerodynamic sideforce is zero, would the "apparent load" vector be exactly "square" to the wingspan. And now-- keeping in mind that the weight vector makes no contribution to the "apparent load" vector-- we understand why the inclinometer ball behaves as it does in slipping, skidding, and coordinated turns. And we've come to this understanding without ever invoking some sort of hypothetical "imbalance" between "centrifugal force" and some other force. It all boils down to the question of whether the airflow is, or is not, striking the side of the fuselage and generating an aerodynamic sideforce vector.)

Because when include the pseudoforce called "centrifugal force" in our vector diagrams, we are using an aircraft-based reference frame rather than an airmass-based reference frame or ground-based reference frame. Since the aircraft can't accelerate relative to itself, the net force in the aircraft-based reference frame will always be zero, whether the aircraft is turning or not. In the aircraft-based reference frame, the fact that the centrifugal force vector exists at all is actually evidence that the aircraft is turning. But for teaching purposes, using the aircraft-based reference frame (and therefore including the "centrifugal force" vector whenever the flight path is not linear) is arguably an inferior approach to using the airmass-based reference frame (and therefore not including any "centrifugal force" vector.)

1. In this answer, when we talk about "horizontal", we mean as seen looking at the aircraft in a head-on view. We're not referring to fore-and-aft forces such as thrust and drag.

2. Note that in the diagrams attached to the question, the illustrator chose to make the "total load" vectors in the "skidding turn" and "coordinated turn" identical, and to make the "HCL" (horizontal component of lift) vectors in the "slipping turn" and "coordinated turn" identical. That's all kind of random and makes no sense.

3. For an extreme case, consider sustained linear "knife-edge" flight at a 90-degree bank angle-- here the aerodynamic sideforce from the airflow striking the side of the fuselage is doing all the work of supporting the aircraft's weight, and the wing's lift vector is zero.