The vectors a=x i-3 j-k and b=2 x i+x j-k include an acute angle and b and positive y-axis include an obtuse angle. Then values of x may be

Question

The vectors a=x i-3 j-k and b=2 x i+x j-k include an acute angle and b and positive y-axis include an obtuse angle. Then values of x may be (A) -2 (B) -3 (C) all x < 0 (D) all x > 0

Tardigrade Admin · Accepted Answer

According to the question a ⋅ b>0 and b ⋅ j<0 ⇒ 2 x2-3 x+1>0 and x<0 ⇒ (2 x-1)(x-1)>0 and x<0 i.e., (x<(1/2). or .x>1) and x<0 ⇒ x<0

Q. The vectors $a = x i - 3 j - k$ and $b = 2 x i + x j - k$ include an acute angle and $b$ and positive $y$ -axis include an obtuse angle. Then values of $x$ may be

$- 2$

$- 3$

all $x < 0$

all $x > 0$

According to the question
$a \cdot b > 0$ and $b \cdot j < 0$
$\Rightarrow 2 x^{2} - 3 x + 1 > 0$ and $x < 0$
$\Rightarrow (2 x - 1) (x - 1) > 0$ and $x < 0$
i.e., $(x < \frac{1}{2}$
or $x > 1)$ and $x < 0$
$\Rightarrow x < 0$

Q. The vectors a=xi−3j−k and b=2xi+xj−k include an acute angle and b and positive y-axis include an obtuse angle. Then values of x may be

−2

−3

all x<0

all x>0