Let $\vec{a}, \vec{b}$ and $\vec{c}$ be vectors of equal magnitude such that the angle between $\vec{a}$ and $\vec{b}$ is $\alpha$,$\vec{b}$ and $\vec{c}$ is $\beta$,and $\vec{c}$ and $\vec{a}$ is $\gamma$. Then the minimum value of $\cos \alpha + \cos \beta + \cos \gamma$ is

Find the sine of the angle between the vectors $\vec{a}=3 \hat{i}+\hat{j}+2 \hat{k}$ and $\vec{b}=2 \hat{i}-2 \hat{j}+4 \hat{k}$.

Let $\bar{a} = \bar{i} + 2\bar{j} + 3\bar{k}$,$\bar{b} = 2\bar{i} - 3\bar{j} + \bar{k}$,and $\bar{c} = 3\bar{i} + \bar{j} - 2\bar{k}$ be three vectors. If $\bar{r}$ is a vector such that $\bar{r} \cdot \bar{a} = 0$,$\bar{r} \cdot \bar{b} = -2$,and $\bar{r} \cdot \bar{c} = 6$,then find the value of $\bar{r} \cdot (3\bar{i} + \bar{j} + \bar{k})$.

If $\vec{a}$ and $\vec{b}$ are two vectors such that $|\vec{a}|=3, |\vec{b}|=4, |\vec{a}+\vec{b}|=\sqrt{37}, |\vec{a}-\vec{b}|=k$ and the angle between $\vec{a}$ and $\vec{b}$ is $\theta$,then find the value of $\frac{4}{13}(k \sin \theta)^2$.

If $|\vec{x}| = |\vec{y}| = |\vec{x} + \vec{y}| = 1$,then $|\vec{x} - \vec{y}| = $ . . . . . . .

If $|\vec{a}|=4, |\vec{b}|=5, |\vec{a}-\vec{b}|=3$ and $\theta$ is the angle between the vectors $\vec{a}$ and $\vec{b}$,then $\cot^2 \theta=$