If $|a|=2$ and $|b|=3$ and the angle between $a$ and $b$ is $120^{\circ}$,then the length of the vector $\left|\frac{a}{2}-\frac{b}{3}\right|$ is

The angle between two consecutive sides $\vec{a}$ and $\vec{b}$ of a parallelogram is $\frac{\pi}{6}$. Given $\vec{a} = (2, -2, 1)$ and $\vec{b} = 2|\vec{a}|$,the area of the parallelogram is . . . . . . sq. units.

If $|a| = 3$,$|b| = 4$ and the angle between $a$ and $b$ is $120^\circ$,then $|4a + 3b| = $

If $(\vec{a}+3 \vec{b})$ is perpendicular to $(7 \vec{a}-5 \vec{b})$ and $(\vec{a}-4 \vec{b})$ is perpendicular to $(7 \vec{a}-2 \vec{b})$,then the angle between $\vec{a}$ and $\vec{b}$ (in degrees) is $......$

If the angle between $\bar{a} = x\bar{i} - 3\bar{j} - \bar{k}$ and $\bar{b} = 2x\bar{i} + x\bar{j} - \bar{k}$ is acute and the angle between $\bar{b}$ and the $y$-axis is obtuse,then $x$ belongs to which interval?

Let $\vec{a}$ be a vector in the plane containing vectors $\vec{b}=\hat{i}+2 \hat{j}+\hat{k}$ and $\vec{c}=2 \hat{i}-\hat{j}+\hat{k}$. If $\vec{a}$ is perpendicular to $\hat{i}+\hat{j}+3 \hat{k}$ and its projection on $\vec{b}$ is $3 \sqrt{6}$, then $|\vec{a}|^2=$