The vectors 2 hati-m hatj+3m hatk, and (1+m) hatj-2m hatj+ hatk include an acute angle for

Question

The vectors 2 hati-m hatj+3m hatk, and (1+m) hatj-2m hatj+ hatk include an acute angle for (A) for values of m (B)  m < - 2 or m > - (1/2) (C) m=-(1/2) (D) m∈[-2,-(1/2)]

Tardigrade Admin · Accepted Answer

The given vectors include an acute angle if (2 hati-m hatj+3m hatk)⋅[(1+m) hati-2m hatj+ hatk] > 0 ⇒ 2(1+m)+2m2+3m > 0 ⇒ 2m2+5m+2 > 0 ⇒ (2m+1)(m+2) > 0 ⇒ m < -2 or m > -(1/2).

Q. The vectors $2 \hat{i} - m \hat{j} + 3 m \hat{k},$ and $(1 + m) \hat{j} - 2 m \hat{j} + \hat{k}$ include an acute angle for

for values of m

$m < - 2 or m > - \frac{1}{2}$

$m = - \frac{1}{2}$

$m \in [- 2, - \frac{1}{2}]$

The given vectors include an acute angle if
$(2 \hat{i} - m \hat{j} + 3 m \hat{k}) \cdot [(1 + m) \hat{i} - 2 m \hat{j} + \hat{k}] > 0$
$\Rightarrow 2 (1 + m) + 2 m^{2} + 3 m > 0$
$\Rightarrow 2 m^{2} + 5 m + 2 > 0$
$\Rightarrow (2 m + 1) (m + 2) > 0$
$\Rightarrow m < - 2$ or $m > - \frac{1}{2}$ .

Q. The vectors 2i^−mj^​+3mk^, and (1+m)j^​−2mj^​+k^ include an acute angle for

for values of m

m<−2orm>−21​

m=−21​

m∈[−2,−21​]

The given vectors include an acute angle if (2i^−mj^​+3mk^)⋅[(1+m)i^−2mj^​+k^]>0 ⇒2(1+m)+2m2+3m>0 ⇒2m2+5m+2>0 ⇒(2m+1)(m+2)>0 ⇒m<−2 or m>−21​.

Q. The vectors $2 \hat{i} - m \hat{j} + 3 m \hat{k},$ and $(1 + m) \hat{j} - 2 m \hat{j} + \hat{k}$ include an acute angle for

$m < - 2 or m > - \frac{1}{2}$

$m = - \frac{1}{2}$

$m \in [- 2, - \frac{1}{2}]$

The given vectors include an acute angle if
$(2 \hat{i} - m \hat{j} + 3 m \hat{k}) \cdot [(1 + m) \hat{i} - 2 m \hat{j} + \hat{k}] > 0$
$\Rightarrow 2 (1 + m) + 2 m^{2} + 3 m > 0$
$\Rightarrow 2 m^{2} + 5 m + 2 > 0$
$\Rightarrow (2 m + 1) (m + 2) > 0$
$\Rightarrow m < - 2$ or $m > - \frac{1}{2}$ .