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

Options

A. ` 1/2,2`
B. ` -2,3`
C. all ` x lt 0`
D. ` x gt 0`

Answer

C

Explanation

Since the angle between the vectors `veca and vecb` is acute and the angle between ` vecb` and y-axis is obtuse. Therefore,
` veca.vecb gt 0 and vecb.hatj lt 0`
` Rightarrow 2x^(2) -3x =1 gt 0 and x lt0`
` Rightarrow ( 2x -1) (x-1) gt 0 and x lt 0`
` Rightarrow ( x lt 1//2 or x gt 1) and x lt 0 Rightarrow x gt 0`