If $\theta$ is the angle between the vectors $\vec{a}$ and $\vec{b}$,then $\frac{|\vec{a} \times \vec{b}|}{|\vec{a} \cdot \vec{b}|}$ 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

Let $\vec{a}$ and $\vec{b}$ be two unit vectors. If $\vec{c} = \vec{a} + 2\vec{b}$ and $\vec{d} = 5\vec{a} - 4\vec{b}$ are perpendicular to each other,then the angle between $\vec{a}$ and $\vec{b}$ is

$\vec{a}, \vec{b}, \vec{c}$ are three unit vectors such that $|\vec{a}+\vec{b}+\vec{c}|=1$ and $\vec{a}$ is perpendicular to $\vec{b}$. If $\vec{c}$ makes angles $\alpha, \beta$ with $\vec{a}, \vec{b}$ respectively,then $\cos \alpha+\cos \beta=$

If $a=\hat{i}+\hat{j}+\hat{k}$,$a \cdot b=1$ and $a \times b=\hat{j}-\hat{k}$,then $b=$

If $|\vec{a}|=4$ and $|\vec{b}|=5$,then the values of $k$ for which $\vec{a}+k \vec{b}$ is perpendicular to $\vec{a}-k \vec{b}$ are