In the problems that I have to apply Grassmann in projective geometry, can I use the inclusion-exclusion principle?
Consider the following problem: We consider three linear varieties of dimension three $V_1$, $V_2$ and $V_3$ in a projective space such that $\dim V_1\cap V_2=\dim V_2\cap V_3=\dim V_3\cap V_1=1$ and $V_1\cap V_2\cap V_3=\emptyset$. Calculate the dimension of the linear variety $V_1\vee V_2\vee V_3$ that they generate.
Mi solution using the inclusion-exclusion principle is: $$\dim(V_1\vee V_2\vee V_3)=\dim V_1+\dim V_2+\dim V_3-\dim(V_1 \cap V_2)-\dim(V_2 \cap V_3)-\dim(V_3 \cap V_1)+\dim(V_1 \cap V_2 \cap V_3)=3+3+3-1-1-1+(-1)=5$$
It is correct and why can I apply this principle?