The component of vector $\overrightarrow{\text{A}}=a_x\hat{i}+a_y\hat{j}+a_z\hat{k}$  along the direction of $\hat{i}-\hat{j}$  is

Let $\overrightarrow{B}=\hat{i}-\hat{j}$

Then component of vector $\overrightarrow{A}$  along $\overrightarrow{B}$

$=\frac{\overrightarrow{A}.\overrightarrow{B}}{\left|\overrightarrow{B}\right|}$

$=\frac{\left(a_x\hat{i}+a_y\hat{j}+a_z\hat{k}\right).\left(\hat{i}-\hat{j}\right)}{\left|\hat{i}-\hat{j}\right|}$

$=\frac{a_x-a_y}{\sqrt{2}}$