Component of $3\hat{i}+4\hat{j}$ perpendicular to $\hat{i}+\hat{j}$ and in the same plane as that of $3\hat{i}+4\hat{j}$ is

The correct option is A $\frac{1}{2}(-\hat{i}+\hat{j})$
We know that component of $\overrightarrow{A}$ perpendicular to $\overrightarrow{B}$ is $\overrightarrow{A}-(\overrightarrow{A}.\overrightarrow{B})\frac{\overrightarrow{B}}{|\overrightarrow{B}{|}^2}$


Component of $3\hat{i}+4\hat{j}$ perpendicular to $(\hat{i}+\hat{j})$ is
$(3\hat{i}+4\hat{j})-\left(\right(3\hat{i}+4\hat{j}).(\hat{i}+\hat{j}\left)\right)\frac{\hat{i}+\hat{j}}{|\hat{i}+\hat{j}{|}^2}=(3\hat{i}+4\hat{j})-7\frac{(\hat{i}+\hat{j})}{{\sqrt{2}}^2}=(3\hat{i}+4\hat{j})-7\frac{(\hat{i}+\hat{j})}{2}=\frac{(-\hat{i}+\hat{j})}{2}$