Question

# The values of $$x$$ for which the angle between the vectors $$\overrightarrow { a } =x\hat { i } -3\hat { j } -\hat { k }$$ and $$\overrightarrow { b } =2x\hat { i } +x\hat { j } -\hat { k }$$ is acute, and the angle between the vectors $$\overrightarrow{b}$$ and the $$y$$-axis of ordinates is obtuse, are

A
1,2
B
2,3
C
All x<0
D
All x>0

Solution

## The correct option is C All $$x<0$$Since the angle between the vectors $$\overrightarrow{a}$$ and $$\overrightarrow{b}$$ is acute and the angle between $$\overrightarrow{b}$$ and the $$y$$-axis is obtuse, therefore, $$\overrightarrow{a}.\overrightarrow{b}>0$$ and $$\overrightarrow{b}.\hat{j}<0$$$$\Rightarrow 2{ x }^{ 2 }-3x+1>0$$ and $$x<0$$$$\Rightarrow \left( 2x-1 \right) \left( x-1 \right) >0$$ and $$x<0$$$$\displaystyle(x<\dfrac{1}{2}$$ or $$x>1)$$ and $$x<0\Rightarrow$$ all $$x<0$$Maths

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