Standard three and two dimensional unit vectors
Theorem
i) Every plane vector can be wrtten as = (a1, a2) = a 1 +a2 and conversely.
ii) Every space vector = (a1,a2,a3) can be written as = (a1,a2,a3) = a1+a2+a3
and conversely.
Proof: i) Here, we have L.H.S. = 
= (a1,0) + (0,a2) = a1(1,0) +a2(0,1) =a 1 +a2
= R.H.S. Again, R.H.S. = a 1 +a2 = a1(1,0) +a2(0,1) = (a1,0) +(0,a2) = (a1,a2) L.H.S. proved.
Similarly we can prove the second part of the theorem also.
= (a1,0) + (0,a2) = a1(1,0) +a2(0,1) =a 1 +a2
= R.H.S. Again, R.H.S. = a 1 +a2 = a1(1,0) +a2(0,1) = (a1,0) +(0,a2) = (a1,a2) L.H.S. proved.
Similarly we can prove the second part of the theorem also.