Inclusion–exclusion principle in projective geometry

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?

Thanks