If $a = i + j + k$,$b = i + 3j + 5k$,and $c = 7i + 9j + 11k$,then the area of the parallelogram having diagonals $a + b$ and $b + c$ is

If the position vectors of three points $A, B, C$ are $\hat{i} + \hat{j} + \hat{k}$,$2\hat{i} + 3\hat{j} - 4\hat{k}$,and $7\hat{i} + 4\hat{j} + 9\hat{k}$ respectively,then find the unit vector perpendicular to the plane of triangle $ABC$.

If non-zero vectors $a$ and $b$ are perpendicular to each other,then what is the solution for $r \times a = b$?

If $\bar{a}=\hat{j}-\hat{k}$ and $\bar{c}=\hat{i}-\hat{j}-\hat{k}$,then the vector $\bar{b}$ satisfying $\bar{a} \times \bar{b}+\bar{c}=\vec{0}$ and $\bar{a} \cdot \bar{b}=3$ is

If $\vec{a}=\hat{i}+\hat{j}+\hat{k}$ and $\vec{b}=2\hat{i}+\hat{j}+\hat{k}$ are two vectors,and $\vec{c}$ is a unit vector lying in the plane of $\vec{a}$ and $\vec{b}$ such that $\vec{c}$ is perpendicular to $\vec{b}$,then find the value of $\vec{c} \cdot (\hat{i}+\hat{j}+2\hat{k})$.

If $a=\hat{i}+\hat{j}+\hat{k}$,$c=\hat{j}-\hat{k}$,$a \times b=c$,and $a \cdot b=3$,then $b=$