The vector component of $\mathrm{A}$ in the direction of $\mathrm{B}$ is $A \cos \theta=\frac{5}{\sqrt{2}}$

Vector $\mathrm{A}=2 \hat{i}+3 \hat{j}$

Vector $\mathrm{B}=\hat{i}+\hat{j}$

We know that,

The vector component of $A$ in the direction of $B$ is

$\vec{A} \cdot \vec{B}=|A||B| \cos \theta$

$A \cos \theta=\frac{\vec{A} \cdot \vec{B}}{|B|}$

$A \cos \theta=\frac{2 \hat{\imath}+3 \hat{\jmath} \cdot \hat{\imath}+\hat{\jmath}}{\sqrt{2}}$

$A \cos \theta=\frac{5}{\sqrt{2}}$

Hence, The vector component of $A$ in the direction of $B$ is $A \cos \theta=\frac{5}{\sqrt{2}}$