The value of x for which the angle between th | vedclass

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(D) Let $\vec{a} = x\hat{i} - 3\hat{j} - \hat{k}$ and $\vec{b} = 2x\hat{i} + x\hat{j} - \hat{k}$.Condition $1$: The angle between $\vec{a}$ and $\vec{

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None of these

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(D) Let $\vec{a} = x\hat{i} - 3\hat{j} - \hat{k}$ and $\vec{b} = 2x\hat{i} + x\hat{j} - \hat{k}$.Condition $1$: The angle between $\vec{a}$ and $\vec{b}$ is acute,so $\vec{a} \cdot \vec{b} > 0$.$\vec{a} \cdot \vec{b} = (x)(2x) + (-3)(x) + (-1)(-1) = 2x^2 - 3x + 1 > 0$.Solving $2x^2 - 3x + 1 > 0$,we get $(2x - 1)(x - 1) > 0$,which implies $x  1$.Condition $2$: The angle between $\vec{b}$ and the $y$-axis (unit vector $\hat{j}$) is obtuse,so $\vec{b} \cdot \hat{j} $\vec{b} \cdot \hat{j} = (2x\hat{i} + x\hat{j} - \hat{k}) \cdot (0\hat{i} + 1\hat{j} + 0\hat{k}) = x Combining both conditions: $x  1$).The intersection is $x Since none of the options match $x < 0$,the correct answer is $D$.

The value of $x$ for which the angle between the vectors $\vec{a} = x\hat{i} - 3\hat{j} - \hat{k}$ and $\vec{b} = 2x\hat{i} + x\hat{j} - \hat{k}$ is acute,and the angle between the vector $\vec{b}$ and the $y$-axis (axis of ordinate) is obtuse,are:

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