Assume that a and b are nonzero vectors in 3 dimensional. Is it ever the cause that projection a on b =projection b on a? If so, exactly when does this occur? The condition should be both necessary and sufficient.

I think that there would be case like that, and I am thinking that can two vectors that made 90 degree work, such as I=(1,0), and j=(0,1), and the projection a on b =projection b on a would = 0? But what would a "0" in projection mean?
Or is there a better example on when does that occur?

Try also with the vector b being a multiple of the vector a, for example, b=2a. What happens then?

Consider the formula for the magnitude of the projection of on : it's . Conversely, the projection of on is . So for the two projections to be equal in magnitude you must have:



Can you think of coditions for either |a| and |b|or angle that make this true?