If $\vec{a}, \vec{b}, \vec{c}$ are unit vectors such that $\vec{a}+\vec{b}+\sqrt{3} \vec{c}=\overrightarrow{0}$,then the angle between $\vec{a}$ and $\vec{b}$ is

Consider three vectors $\overrightarrow{a}, \overrightarrow{b}, \overrightarrow{c}$. Let $|\overrightarrow{a}|=2, |\overrightarrow{b}|=3$ and $\overrightarrow{a}=\overrightarrow{b} \times \overrightarrow{c}$. If $\alpha \in [0, \frac{\pi}{3}]$ is the angle between the vectors $\overrightarrow{b}$ and $\overrightarrow{c}$,then the minimum value of $27|\overrightarrow{c}-\overrightarrow{a}|^2$ is equal to :

Let $\vec{a} = \lambda \hat{i} + \hat{j} + \hat{k}$, $\vec{b} = 2\hat{i} + 4\hat{j} + 4\hat{k}$, and $\vec{c} = \hat{i} + \mu \hat{j} + \hat{k}$. If $\vec{a}$ is parallel to $\vec{b}$ and $\vec{b}$ is perpendicular to $\vec{c}$, then find the value of $\lambda - \mu$.

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 $\theta$ is the angle between vectors $\vec{a}$ and $\vec{b}$ and $|\vec{a} \times \vec{b}| = |\vec{a} \cdot \vec{b}|$, then $\theta$ is equal to

The angle between the vectors $2 \hat{k} - 3 \hat{j}$ and $\hat{i} - 2 \hat{k}$ is