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

Find the distance between the lines $l_{1}$ and $l_{2}$ given by $\vec{r}=\hat{i}+2 \hat{j}-4 \hat{k}+\lambda(2 \hat{i}+3 \hat{j}+6 \hat{k})$ and $\vec{r}=3 \hat{i}+3 \hat{j}-5 \hat{k}+\mu(2 \hat{i}+3 \hat{j}+6 \hat{k})$.

If $|\vec{a}| = \sqrt{27}$,$|\vec{b}| = 7$ and $|\vec{a} \times \vec{b}| = 35$,then $\vec{a} \cdot \vec{b}$ is equal to

Let $\overrightarrow{a} = 2\hat{i} - 3\hat{j} + 4\hat{k}$ and $\overrightarrow{b} = 7\hat{i} + \hat{j} - 6\hat{k}$. If $\overrightarrow{r} \times \overrightarrow{a} = \overrightarrow{r} \times \overrightarrow{b}$ and $\overrightarrow{r} \cdot (\hat{i} + 2\hat{j} + \hat{k}) = -3$,then $\overrightarrow{r} \cdot (2\hat{i} - 3\hat{j} + \hat{k})$ is equal to:

If $\overline{a}=\hat{i}+\hat{j}$,$\overline{b}=2\hat{j}-\hat{k}$ and $\overline{r} \times \overline{a}=\overline{b} \times \overline{a}$,$\overline{r} \times \overline{b}=\overline{a} \times \overline{b}$,then the value of $\frac{\overline{r}}{|\overline{r}|}$ is

For what value of $m$ is the angle between the vectors $2\bar{i} - m\bar{j} + 3m\bar{k}$ and $(1 + m)\bar{i} - 2m\bar{j} + \bar{k}$ acute?