Let ` vec a=x^2 hat i+2 hat j-2 hat k ,\ vec b= hat i- hat j+ hat k\ a n d\ vec c=x^2 hat i+5 hat j-4 hat k` be three vecrors find the values of `x` for which the angle between ` vec a\ a n d\ vec b` acute and the angle between ` vec b\ a n d\ vec c` is obtuse.

Explanation

Given that,
` vec a=x^2 hat i+2 hat j-2 hat k` ,
`vec b= hat i- hat j+ hat k`
a n d ` vec c=x^2 hat i+5 hat j-4 hat k`
` vec a* vec b=|vec a||vec b|cos theta`
according to question,
the angle between ` vec a\ a n d\ vec b`
acute and the angle between ` vec b\ a n d\ vec c`
is obtuse.
so, `vec a * vec b gt 0`
`vec b* vec c lt 0`
`vec a * vec b gt 0`
`=>x^2-2-2 gt 0`
`=>(x-2)(x+2) gt 0`
`=> x in (-oo,-2) uu (2,oo)`
again ,
`vec a * vec b gt 0`
`=>x^2-5-4 lt 0`
`=>x^2-9 lt 0`
`=>(x-3)(x+3) lt 0`
`=>x in (-3,3)`
hence `x in (-3,2) uu (2,3)`