# Why are displacements added algebraically during interference of waves , why not vectorially?

In the diagram (A) , the individual waves are crossing at an angle to each other , so the dispacemnt of the particle at their intersection must have an angle between them , and that must involve vector addition and not algebraic addition , The direction of displacemnt of the particle will be different from the direction of displacements of individual waves.Why are the displacements being added algebraically then , I am having a Resnick Halliday Walker , I have included the pictures of the pages concerned from two different chapters , the equation from chapter 16 was used in 17. While in 16 the equation was derived for two travelling in one direction ,i.e not inclined mutually , while it was used in chapter 17 for two waves travelling inclined to each other to my surprise.

I think I know the problem , could it be that the diagram A that I drew is wrong ,and the two wave are not flat on the screen but in a perpendicular plane to the screen , that way algebraic addition works , if that's the case , then how to find the equation for resultant wave in the configuration I have setup in diagram A

You are correct: the fields are added vectorially in every case. However, we almost always work in the far field approximation. In this limit the two lines $$\ell_1$$ and $$\ell_2$$ you drew are almost parallel, so the displacement fields (or electric fields, etc., demanding on the situation) are in almost the same direction, so they effectively add algebraically.