# Which of the following shows correct triangle law of vector addition? [closed]

This here shows an aerofoil of an axial flow compressor with velocities $$V_a$$ = air velocity and $$V_e$$ = velocity of blade.The resultant or relative velocity with which air strikes the aerofoil is $$V_r$$ . As per what I have learned Fig (ii) is the correct option. but some texts show these. And in this video fig at 11:44 of video the resultant velocity is as per fig (i).

So I want to know if there is any mistake in the way that I understood triangle law (which is if you have two vectors and if you join them with head of one resting on the tail of other then the arrow joining the free ends give resultant). Or is there any other basics that I am missing because the resultant in fig (i) and fig (ii) are in different directions.

Your v$$_e$$ is wrong. It shows the speed of the blade but should point in the opposite direction so it shows the speed of the air approaching the blade.

Then of course diagram (i) is correct.

• I have thought of the possibility that blade is rotating in the wrong direction in that case if we reverse the velocity of rotor blade then fig(i) is right.But doesn't the fig 'a' in the image shows a similar scenario with rotor blade rotating in same direction as i drew and with aerofoil oriented in same way.Does that mean the velocity triangle in text is wrong? Apr 25, 2020 at 14:38
• @AbhishekPallipparagopakumar. Yes, it does. The author made the same mistake. Figure yourself sitting on the nose of that airfoil and watch how the wind is blowing towards you. Both the wind from axial flow and from radial movement must be represented in the same way. That is not the case in both speed triangles in Fig. 9.26. Apr 25, 2020 at 14:45
• So you are saying that author should have shown the relative air velocity due to rotation of blade instead of showing the velocity of the blade. Apr 25, 2020 at 15:00
• This answer and the attached comments are misleading. You are free to define your own vectors to describe the air flow, but just because you do this differently from the text does not make the text wrong. Apr 25, 2020 at 15:30
• @hiccups Consistency is important here. One vector denotes the movement of the air, the other the movement of the blade. It does not make sense to add them. Or, if they must be added, put them in the same reference system. That is not misleading at all. Apr 25, 2020 at 17:15

If $$\boldsymbol{V}_r=\boldsymbol{V}_a+\boldsymbol{V}_e$$ then yes, your diagram (ii) would be correct. You are not missing any basics in vector addition.

Edit: Since the source of confusion in the question is not really that profound, I was hoping that my initial answer would provide enough of a hint to resolve it. But as I have already elaborated on the problem in my comments, I may as well also state it explicitly in my answer for completeness. Below, I will use the notation from the text excerpt provided: $$\boldsymbol{C}_1=\boldsymbol{V}_a$$, $$\boldsymbol{U}=\boldsymbol{V}_e$$, and $$\boldsymbol{W}_1=\boldsymbol{V}_r$$.

Given my hint above, the reader is presented with two statements:

1. If $$\boldsymbol{W}_1=\boldsymbol{C}_1+\boldsymbol{U}$$ then the diagram (ii) in the question is correct.
2. The vector triangle in the text excerpt corresponds to diagram (i) in the question.

For the two statements to be consistent, then, implies$$^\dagger$$ that the condition of statement 1 is not true. In other words, $$\begin{equation} \boldsymbol{W}_1\neq\boldsymbol{C}_1+\boldsymbol{U}\,. \end{equation}$$ Indeed, $$\boldsymbol{W}_1$$ is described as the relative velocity of the incident air w.r.t. the rotor. As with any other relative quantity, it corresponds to a difference of quantities: $$\begin{equation} \boldsymbol{W}_1=\boldsymbol{C}_1-\boldsymbol{U}\,. \end{equation}$$ The vector triangles presented in the text excerpt is consistent with this relationship.

The moral of the story is: If you see a vector triangle... don't arbitrarily choose one of the vectors and assume that it is a sum of the other vectors! Instead, understand what each vector represents, and construct a sensible relationship between them.

$$^\dagger$$Well, alternatively the text could be wrong. But one should always explore other possibilities first, since more often than not, the reader is the novice while the author is the subject matter expert.

• So does that mean the vector diagram in the fig and video i have shown is wrong? Apr 25, 2020 at 11:02
• This is not about mindless addition of arrows but getting the arrows right in the first place. Apr 25, 2020 at 14:15
• @PeterKämpf On the contrary: the confusion is entirely to do with the mindless "addition of arrows". Or, more specifically, why one shouldn't mindlessly assume addition of vectors. There is nothing mistaken about how the text defines its vectors. Apr 25, 2020 at 15:25