Determine the value of `c` so that for all real `x` , vectors `c x hat i-6 hat j-3 hat k and x hat i+2 hat j+2c x hat k` make an obtuse angle with each other.

Answer

`-4//3lt c lt 0`

Explanation

We have , `veca = cx hati - 6hatj - 3hatk`
`vecb = xhati + 2hatj + 2 2cx hatk`
now we know that `veca.vecb = |veca||vecb|cos theta`
As the angle between `veca and vecb` is obtuse, `cos theta lt 0`
`Rightarrow veca .vecb lt 0`
`Rightarrow cx^(2)-12 + 6cx lt 0`
` Rightarrow -cx^(2) - 6cx +12 gt 0, x in R`
`Rightarrow -c gt and D gt 0`
` Rightarrow c gt 0 36c^(2) + 48c lt0`
` Rightarrow c lt 0 and ( 3 c + 4) lt 0`
` Rightarrow c lt 0 and c gt - 4//3`
` Rightarrow -4//3 lt c lt 0`