If $a=2 \hat{i}+\hat{k}$,$b=\hat{i}+\hat{j}+\hat{k}$,and $c=4 \hat{i}-3 \hat{j}+7 \hat{k}$,then the vector $r$ satisfying $r \times b=c \times b$ and $r \cdot a=0$ is

Let $\overline{a}=3 \hat{i}-\alpha \hat{j}+\hat{k}$ and $\overline{b}=\hat{i}+\alpha \hat{j}+3 \hat{k}$. If the area of the parallelogram whose adjacent sides are represented by the vectors $\overline{a}$ and $\overline{b}$ is $8 \sqrt{3}$ sq. units,then $\overline{a} \cdot \overline{b}$ is equal to

Let $a, b, c$ be three vectors such that $a \neq 0$,$a \times b = 2a \times c$,$|a| = |c| = 1$,$|b| = 4$,and $|b \times c| = \sqrt{15}$. If $b - 2c = \lambda a$,then $\lambda$ equals:

If $\theta$ is the angle between the vectors $2 \hat{i}-\hat{j}+2 \hat{k}$ and $a \hat{i}+4 \hat{j}+b \hat{k}$ and $\cos \theta=\frac{2}{3}$,then $2(a+b+3)=$

$\hat{i} \cdot (\hat{k} \times \hat{j}) + \hat{j} \cdot (\hat{i} \times \hat{k}) + \hat{k} \cdot (\hat{i} \times \hat{j}) = \_\_\_\_$

Consider three vectors $\overrightarrow{a}, \overrightarrow{b}, \overrightarrow{c}$. Let $|\overrightarrow{a}|=2, |\overrightarrow{b}|=3$ and $\overrightarrow{a}=\overrightarrow{b} \times \overrightarrow{c}$. If $\alpha \in [0, \frac{\pi}{3}]$ is the angle between the vectors $\overrightarrow{b}$ and $\overrightarrow{c}$,then the minimum value of $27|\overrightarrow{c}-\overrightarrow{a}|^2$ is equal to :