Question

Two Electric field vectors E1 and E2 are perpendicular to each other; obtain its base
vectors.

Answer 1

Imagine that these two vectors make up a resulting vector E θ degrees above i-axis, therefore, we can write that

$$\vec{E}=|E|\text{cos}\theta \textbf{i}+|E|\text{sin}\theta \textbf{j}.$$

Now put that

$$a=\frac{|E_2|}{|E|\text{cos}\theta},\space b=\frac{|E_1|}{|E|\text{sin}\theta}.$$

Thus,

$$\vec{E}=a|E_2|\textbf{i}+b|E_1|\textbf{j}.$$

Now the vector E can be written as a linear combination of $\vec{E_1},\vec{E_2}.$ The basis vector is

$$<\vec{E_1},\vec{E_2}>.$$