For what value of $x$ is the angle between the vectors $\vec{a} = -3\hat{i} + x\hat{j} + \hat{k}$ and $\vec{b} = x\hat{i} + 2x\hat{j} + \hat{k}$ acute,and the angle between $\vec{b}$ and the $x$-axis lies between $\pi/2$ and $\pi$?

If $a \cdot b = 0$,then:

If $\vec{a} = \hat{i} + 3\hat{j} - 2\hat{k}$ and $\vec{b} = 4\hat{i} - 2\hat{j} + 4\hat{k}$,then $(2\vec{a} + \vec{b}) \cdot (\vec{a} - 2\vec{b}) = \dots$

If $A=(-2,2,3), B=(3,2,2), C=(4,-3,5)$ and $D=(7,-5,-1)$,then the projection of $\overline{AB}$ on $\overline{CD}$ is

Let $\vec{a}, \vec{b}, \vec{c}$ be $3$ vectors such that $|\vec{a}|=3, |\vec{b}|=2\sqrt{2}, |\vec{c}|=5$ and $\vec{c}$ is perpendicular to the plane of $\vec{a}$ and $\vec{b}$. If the angle between the vectors $\vec{a}$ and $\vec{b}$ is $\frac{\pi}{4}$, then $|\vec{a}+\vec{b}+\vec{c}|=$

If $a$ and $b$ are two unit vectors and $\theta$ is the angle between them,then the unit vector along the angular bisector of $a$ and $b$ is given by