Given that $straight a with rightwards arrow on top. space straight b with rightwards arrow on top space equals space 0 space space and space straight a with rightwards arrow on top space cross times space straight b with rightwards arrow on top space equals space 0 with rightwards arrow on top.$ what can you conclude about the vectors $straight a with rightwards arrow on top space and space straight b with rightwards arrow on top$ ?

Solution

We have
$straight a with rightwards arrow on top. space straight b with rightwards arrow on top space equals space 0$ ...(1)
and $straight a with rightwards arrow on top space cross times space straight b with rightwards arrow on top space equals space 0 with rightwards arrow on top$ ...(2)
From (1), it is clear that
either $straight a with rightwards arrow on top space equals space 0 with rightwards arrow on top space space or space space straight b with rightwards arrow on top space equals space 0 with rightwards arrow on top space space space or space space straight a with rightwards arrow on top comma space straight b with rightwards arrow on top space are space perpendicular.$
From (2), it is clear that
either $straight a with rightwards arrow on top space equals space 0 space space space space space or space space space straight b with rightwards arrow on top space equals space 0 with rightwards arrow on top space space space or space space straight a with rightwards arrow on top comma space straight b with rightwards arrow on top space space are space parallel.$
Now, $straight a with rightwards arrow on top comma space straight b with rightwards arrow on top$ are perpendicular and $straight a with rightwards arrow on top space straight b with rightwards arrow on top$ are parallel cannot hold simultaneously.
$therefore$ either $straight a with rightwards arrow on top space equals 0 with rightwards arrow on top space space space or space space straight b with rightwards arrow on top space equals 0 with rightwards arrow on top.$