No, the vector $(\hat{i}+\hat{j})$ is not a unit vector because its magnitude is, \[|\hat{i}+\hat{j}|=\sqrt{|i|^2+|j|^2+2|i||j|\cos90°}\] \[=\sqrt{1^2+1^2+0}=\sqrt{2}\] Since its magnitude is equal to $\sqrt{2}$ not $1$, it is not a unit vector.

The magnitude of a vector has doubled, its direction remaining the same. Can you conclude that the magnitude of each component of the vector has doubled?

Show that two vectors of different magnitudes cannot be combined to give zero resultant, whereas three vectors can be.