The values of x for which the angle between the vectors ` veca =xhati - 3hatj-hatk and vecb = 2x hati + x hatj -hatk` is acute, and the angle, between the vector `vecb` and the axis of ordinates is obtuse, are

Answer

2

Explanation

Let `a=xhati-3hatj-hatk and b=2xhati+xhatj-hatk` be the adjacent sides of the parallelogram.
Now angle between a and b is acute, i.e., `|a+b|gt|a-b|`
`implies|3xhati+(x-3)hatj-2hatk|^(2) gt |-xhati-(x+3)hatj|^(2)`
or `9x^(2)+(x-3)^(2)+4 gt x^(2)+(x+3)^(3)`

or `8x^(2)-12x+4 gt0` or `2x^(2)-3x+1 gt0`
or `(2x-1)(x-1) gt 0 implies x lt (1)/(2) ` or `x gt 1`
Hence, the least positive integral value is 2.