What is the component of $(3\hat{i}+4\hat{j})$ along $(\hat{i}+\hat{j})$?

The correct option is D $\frac{7}{2}(\hat{j}+\hat{i})$

$\text{Step 1: Scalar projection of}\overrightarrow{B}$ $\text{along}$ $\overrightarrow{A}$ $\text{[Ref. Fig.]}$

Let say, $\overrightarrow{B}=3\hat{i}+4\hat{j},$ $\overrightarrow{A}=\hat{i}+\hat{j}$

Magnitude of component of $\overrightarrow{B}$ along $\overrightarrow{A}$$=|\overrightarrow{B}|\cos \theta$

$=\frac{\overrightarrow{A}.\overrightarrow{B}}{|\overrightarrow{A}|}=\frac{3+4}{\sqrt{2}}=\frac{7}{\sqrt{2}}$ $[\because \overrightarrow{A}.\overrightarrow{B}=|\overrightarrow{A}||\overrightarrow{B}|\cos \theta ]$

This is the magnitude of component of $\overrightarrow{B}$ along $\overrightarrow{A}$

$\text{Step 2: Vector Projection}$

Component of $\overrightarrow{B}$ along $\overrightarrow{A}$$=\frac{\overrightarrow{A}.\overrightarrow{B}}{|\overrightarrow{A}|}\hat{A}=\frac{7}{\sqrt{2}}\times \frac{(\hat{i}+\hat{j})}{\sqrt{2}}=\frac{7}{2}(\hat{i}+\hat{j})$

This is the component of $\overrightarrow{B}$ along $\overrightarrow{A}$

Hence option $D$ is correct.