If $a = 2i + 2j - k$ and $b = 6i - 3j + 2k,$ then the value of $a \times b$ is

$\vec{r}$ is a vector perpendicular to the plane determined by the vectors $2 \hat{i}-\hat{j}$ and $\hat{j}+2 \hat{k}$. If the magnitude of the projection of $\vec{r}$ on the vector $2 \hat{i}+\hat{j}+2 \hat{k}$ is $1$, then $|\vec{r}|=$

If $\hat{u}$ and $\hat{v}$ are unit vectors and $\theta$ is the acute angle between them,then for what value of $\theta$ is $2\hat{u} \times 3\hat{v}$ a unit vector?

If $\overline{a}=3 \hat{i}-5 \hat{j}$ and $\overline{b}=6 \hat{i}-3 \hat{j}$ are two vectors and $\overline{c}$ is a vector such that $\overline{c}=\overline{a} \times \overline{b}$,then the ratio $a: b: c$ is:

$A$ unit vector coplanar with $\vec{a} = \hat{i} + \hat{j} + 2\hat{k}$ and $\vec{b} = \hat{i} + 2\hat{j} + \hat{k}$ and perpendicular to $\vec{c} = \hat{i} + \hat{j} + \hat{k}$ is

Let $\overline{a}=\hat{i}+2 \hat{j}-\hat{k}$ and $\overline{b}=\hat{i}+\hat{j}-\hat{k}$ be two vectors. If $\overline{c}$ is a vector such that $\overline{b} \times \overline{c}=\overline{b} \times \overline{a}$ and $\overline{c} \cdot \overline{a}=0$,then $\overline{c} \cdot \overline{b}$ is