Let S be the set of all a ∈ R for which the angle between the vectors u = a ( log e b ) hat i -6 hat j +3 hat k and vecv=( log e b) hati+2 hatj+2 a( log e b) hatk,(b1) is acute. Then S is equal to

Question

Let S be the set of all a ∈ R for which the angle between the vectors u = a ( log e b ) hat i -6 hat j +3 hat k and vecv=( log e b) hati+2 hatj+2 a( log e b) hatk,(b>1) is acute. Then S is equal to (A) (-∞,-(4/3)) (B) Φ (C) (-(4/3), 0) (D) ((12/7), ∞)

Tardigrade Admin · Accepted Answer

For angle to be acute u ⋅ v >0 ⇒ a ( log e b )2-12+6 a ( log e b )>0 ∀ b >1 text let log e b = t ⇒ t >0 text as b >1 y = at 2+6 at -12 y >0, ∀ t >0 ⇒ a ∈ φ

Q. Let $S$ be the set of all $a \in R$ for which the angle between the vectors $u = a (lo g_{e} b) \hat{i} - 6 \hat{j} + 3 \hat{k}$ and $v = (lo g_{e} b) \hat{i} + 2 \hat{j} + 2 a (lo g_{e} b) \hat{k}, (b > 1)$ is acute. Then $S$ is equal to

$(- \infty, - \frac{4}{3})$

$Φ$

$(- \frac{4}{3}, 0)$

$(\frac{12}{7}, \infty)$

For angle to be acute
$u \cdot v > 0$
$\Rightarrow a (lo g_{e} b)^{2} - 12 + 6 a (lo g_{e} b) > 0$
$\forall b > 1$
$let lo g_{e} b = t \Rightarrow t > 0 as b > 1$
$y = a t^{2} + 6 a t - 12& y > 0, \forall t > 0$
$\Rightarrow a \in ϕ$

Q. Let S be the set of all a∈R for which the angle between the vectors u=a(loge​b)i^−6j^​+3k^ and v=(loge​b)i^+2j^​+2a(loge​b)k^,(b>1) is acute. Then S is equal to

(−∞,−34​)

Φ

(−34​,0)

(712​,∞)

For angle to be acute u⋅v>0 ⇒a(loge​b)2−12+6a(loge​b)>0 ∀b>1 let loge​b=t⇒t>0 as b>1 y=at2+6at−12&y>0,∀t>0 ⇒a∈ϕ

Q. Let $S$ be the set of all $a \in R$ for which the angle between the vectors $u = a (lo g_{e} b) \hat{i} - 6 \hat{j} + 3 \hat{k}$ and $v = (lo g_{e} b) \hat{i} + 2 \hat{j} + 2 a (lo g_{e} b) \hat{k}, (b > 1)$ is acute. Then $S$ is equal to

$(- \infty, - \frac{4}{3})$

$Φ$

$(- \frac{4}{3}, 0)$

$(\frac{12}{7}, \infty)$

For angle to be acute
$u \cdot v > 0$
$\Rightarrow a (lo g_{e} b)^{2} - 12 + 6 a (lo g_{e} b) > 0$
$\forall b > 1$
$let lo g_{e} b = t \Rightarrow t > 0 as b > 1$
$y = a t^{2} + 6 a t - 12& y > 0, \forall t > 0$
$\Rightarrow a \in ϕ$