The number of unit vectors perpendicular to $\overline{a}=(1,1,0)$ and $\overline{b}=(0,1,1)$ is

$a=3 \hat{i}+\hat{j}-\hat{k}, b=\hat{i}-4 \hat{j}+5 \hat{k}, c=4 \hat{i}+5 \hat{j}-\hat{k}$ are three vectors and a vector $r$ is perpendicular to both the vectors $b$ and $c$. If $r \cdot a=9$,then $r=$

$A$ unit vector perpendicular to the plane of $a = 2i - 6j - 3k$ and $b = 4i + 3j - k$ is

Let $\vec a = 2\hat i + \hat j - 2\hat k$ and $\vec b = \hat i + \hat j$. If $\vec c$ is a vector such that $\vec a \cdot \vec c + 2|\vec c| = 0$ and $|\vec c - \vec a| = \sqrt{14}$,and the angle between $\vec a \times \vec b$ and $\vec c$ is $30^o$,then $|(\vec a \times \vec b) \times \vec c|$ is:

Find a unit vector perpendicular to both vectors $3i + 2j - k$ and $12i + 5j - 5k$.

If the vector $\vec{b} = 3\hat{j} + 4\hat{k}$ is written as the sum of a vector $\vec{b_1}$,parallel to $\vec{a} = \hat{i} + \hat{j}$ and a vector $\vec{b_2}$,perpendicular to $\vec{a}$,then $\vec{b_1} \times \vec{b_2}$ is equal to