Find the angle between the lines whose direction ratios are $a, b, c$ and $b-c, c-a, a-b$. (in $^{\circ}$)

The constant value $(\lambda + \mu)$ for which the lines $\vec{r} = (2\hat{i} + \hat{j} + \hat{k}) + \lambda(\hat{i} - 2\hat{j})$ and $\vec{r} = (\hat{i} + \hat{j} - 3\hat{k}) + \mu(\hat{j} + 2\hat{k})$ intersect each other is equal to (where $\lambda$ and $\mu$ are parameters).

The value of $\lambda$ for which the vectors $2\lambda \hat{i} + \hat{j} - \hat{k}$ and $2\hat{j} + \hat{k}$ are perpendicular is:

Let $p$ and $q$ be the position vectors of $P$ and $Q$ respectively with respect to $O$ and $|p| = p, |q| = q.$ The points $R$ and $S$ divide $PQ$ internally and externally in the ratio $2 : 3$ respectively. If $\overrightarrow{OR}$ and $\overrightarrow{OS}$ are perpendicular,then:

If $\overrightarrow{a} \cdot \overrightarrow{b} = -|\overrightarrow{a}||\overrightarrow{b}|$,then the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$ is (in $^{\circ}$)

Let the vectors $\vec{a} = -\hat{i} + \hat{j} + 3\hat{k}$ and $\vec{b} = \hat{i} + 3\hat{j} + \hat{k}$. For some $\lambda, \mu \in \mathbb{R}$, let $\vec{c} = \lambda \vec{a} + \mu \vec{b}$. If $\vec{c} \cdot (3\hat{i} - 6\hat{j} + 2\hat{k}) = 10$ and $\vec{c} \cdot (\hat{i} + \hat{j} + \hat{k}) = -2$, then $|\vec{c}|^2$ is equal to: