Question

The values of $x$ for which the angle between the vectors $a=xi-3j-k$ and $b=2xi+xj-k$ is acute, and the angle between the vector $b$ and the axis of ordinates is obtuse, are

• A. $1,2$
• B. $-2,-3$
• C. all $x<0$
• D. all $x>0$

Answer 1

Answer: (B), (C)

The correct options are
B $-2,-3$
C all $x<0$

According to the given conditions, $a\cdot b>0$ and $b\cdot c<0,$ where $c=(0,1,0)$ as c is on the ordinate axis which means x and z are 0
Thus, $2x^2-3x+1>0\Rightarrow (x-1)(2x-1)>0\Rightarrow x\in (-\infty ,1/2)\cup (1,\infty )$ and $x<0$
Combining both we get $x<0$

Hence, $x<0$ is correct. Furthermore, -2 and -3 are both $<0$ which means it is correct too.