If $\overrightarrow{A}=\hat{i}-2\hat{j} , \overrightarrow{B} =2\hat{i}+3\hat{k} and \overrightarrow{C} =\hat{i} +\hat{j}$ ,find component of vector $\mathrm{A}+\mathrm{B}$ along C.

$\mathbf{A}+\mathbf{B}=(\hat{\mathbf{i}}-2\hat{\mathbf{j}})+(2\hat{\mathbf{i}}+3\hat{\mathbf{k}})=3\hat{\mathbf{i}}-2\hat{\mathbf{j}}+3\hat{\mathbf{k}}$

Component of $\mathrm{A}+\mathrm{B}=\mathrm{R}$ (say) along $\mathrm{C}$ is 

$R\cos \theta =\frac{\mathbf{R}·\mathbf{C}}{C}$

$=\frac{(3\hat{i}-2\hat{j}+3\hat{k})·(\hat{i}+\hat{j})}{\sqrt{(1{)}^2+(1{)}^2}}$

$=\frac{3-2}{\sqrt{2}}=\frac{1}{\sqrt{2}}$