If $\theta$ is the angle between two vectors $\vec{a}$ and $\vec{b},$ then $\vec{a} \cdot \vec{b} \ge 0$ if

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

Find the angle between the following pairs of lines:
$\vec{r}=2 \hat{i}-5 \hat{j}+\hat{k}+\lambda(3 \hat{i}+2 \hat{j}+6 \hat{k})$ and
$\vec{r}=7 \hat{i}-6 \hat{k}+\mu(\hat{i}+2 \hat{j}+2 \hat{k})$

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 $\bar{r} \cdot (3\bar{i} + \bar{j} + \bar{k})$.

Let $\bar{a} = \bar{i} + 2\bar{j} + 2\bar{k}$ and $\bar{b} = 2\bar{i} - \bar{j} + p\bar{k}$ be two vectors. If the angle between $\bar{a}$ and $\bar{b}$ is $60^{\circ}$, then $p =$

The horizontal force and the force inclined at an angle $60^\circ$ with the vertical,whose resultant is in the vertical direction with a magnitude of $P \ kg$,are: