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Q.
The vectors $2\,\hat{i}-m\,\hat{j}+3m\,\hat{k}, \, $ and $(1+m)\hat{j}-2m\hat{j}+\hat{k} $ include an acute angle for
Vector Algebra
Solution:
The given vectors include an acute angle if
$\left(2\,\hat{i}-m\,\hat{j}+3m\,\hat{k}\right)\cdot\left[\left(1+m\right)\,\hat{i}-2m\,\hat{j}+\hat{k}\right] > 0$
$\Rightarrow 2\left(1+m\right)+2m^{2}+3m > 0$
$\Rightarrow 2m^{2}+5m+2 > 0$
$\Rightarrow \left(2m+1\right)\left(m+2\right) > 0$
$\Rightarrow m < -2$ or $m > -\frac{1}{2}$.