Answer
$a=\frac{2}{3}$.
Work Step by Step
By definition the dot product of 2 vector $ai+bj$ and $ci+dj$ is: $(ai+bj)(ci+dj)=ac+bd.$ If this product is 0, then the vectors are orthogonal (perpendicular) and hence the enclosed angle is $90^\circ.$
Hence here $v\cdot w=(i-aj)(2i+3j)=1\cdot2+(-a)\cdot3=0.$
We get the equation: $2-3a=0$, hence $a=\frac{2}{3}$.
