The set of all $\alpha$,for which the vectors $\vec{a}=\alpha t \hat{i}+6 \hat{j}-3 \hat{k}$ and $\vec{b}=t \hat{i}-2 \hat{j}-2 \alpha t \hat{k}$ are inclined at an obtuse angle for all $t \in R$ 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:

Let $p$ and $q$ be the position vectors of $P$ and $Q$ respectively with respect to $O$ and $|p| = p, |q| = q.$ The points $R$ and $S$ divide $PQ$ internally and externally in the ratio $2 : 3$ respectively. If $\overrightarrow{OR}$ and $\overrightarrow{OS}$ are perpendicular,then:

$A$ vector coplanar with vectors $i + j$ and $j + k$ and parallel to the vector $2i - 2j - 4k$ is:

If vectors $\bar{a}=2 \hat{i}+2 \hat{j}+3 \hat{k}$,$\bar{b}=-\hat{i}+2 \hat{j}+\hat{k}$,and $\bar{c}=-3 \hat{i}+\hat{j}+2 \hat{k}$ are such that $\bar{a}+\lambda \bar{b}$ is perpendicular to $\bar{c}$,then $\lambda=$

If $\overrightarrow{a}=\hat{i}-\hat{j}-\hat{k}$ and $\overrightarrow{b}=\lambda \hat{i}-3 \hat{j}+\hat{k}$ and the orthogonal projection of $\overrightarrow{b}$ on $\overrightarrow{a}$ is $\frac{4}{3}(\hat{i}-\hat{j}-\hat{k})$,then $\lambda$ is equal to