If a.i=4, then ,what is the value of (axj).(2j-3k) , where a is a vector

Question

This is a question I saw in a question paper of a competitive exam but I was unable to solve it. Can anyone please assist me with any sort of hint to solve this problem and any type of explanation if needed in the hint. Any help will be highly appreciated.

Answer 1

$a\cdot i=4$ means that $a=4i+sj+tk$ where $s$ and $t$ are real, and so $a\times j=4k-ti$. Now take the dot product with $2j-3k$; I think that pesky $t$ will obligingly vanish.

Answer 2

$$(a\times\mathbf{j})\cdot(2\mathbf{j}-3\mathbf{k}) = a\cdot(\mathbf{j}\times (2\mathbf{j}-3\mathbf{k}) = a\cdot(-3\mathbf{i}) = (4\mathbf{i})\cdot (-3\mathbf{i}) = -12$$