The value of $m \in R$,when the angle between the vectors $\bar{p} = m \hat{i} - 6 \hat{j} + 3 \hat{k}$ and $\bar{q} = \hat{i} + 2 \hat{j} + 2m \hat{k}$ is obtuse,is

Let $(p \wedge q)$ denote the angle between vectors $p$ and $q$. If $a + b + c = 0$, $|a| = 7$, $|b| = 5$, and $|c| = 3$, find the correct relation.

$ABCD$ is a quadrilateral with $\overline{AB}=\bar{a}$,$\overline{AD}=\bar{b}$ and $\overline{AC}=2\bar{a}+3\bar{b}$. If its area is $\alpha$ times the area of the parallelogram with $AB$ and $AD$ as adjacent sides,then the value of $\alpha$ is

If $A = (k, 1, -1)$,$B = (2k, 0, 2)$,and $C = (2 + 2k, k, 1)$,and $AB \perp BC$,then the value of $k$ is:

If $4\hat{i}+\hat{j}-\hat{k}$ and $3\hat{i}+m\hat{j}+2\hat{k}$ are at a right angle,then $m = $

If the vectors $\hat{i}+3 \hat{j}+4 \hat{k}$ and $\lambda \hat{i}-4 \hat{j}+\hat{k}$ are orthogonal to each other,then $\lambda$ is equal to