If the vectors $\mathbf{B C}=2 \hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}}$ and $\mathbf{C D}=\hat{\mathbf{i}}+2 \hat{\mathbf{j}}-2 \hat{\mathbf{k}}$ represent two adjacent sides of a parallelogram ABCD and $\theta$ is the angle between its diagonals $\mathbf{A C}$ and $\mathbf{B D}$, then $\tan \theta=$

Let $\mathbf{a}=\lambda \hat{\mathbf{i}}+3 \hat{\mathbf{j}}+4 \hat{\mathbf{k}}, \mathbf{b}=3 \hat{\mathbf{i}}-\hat{\mathbf{j}}+\lambda \hat{\mathbf{k}}$ and $\mathbf{c}=\lambda \hat{\mathbf{i}}+\hat{\mathbf{j}}-3 \hat{\mathbf{k}}$ be three vectors for some integer $\lambda$. If the volume of the parallelopiped with $\mathbf{a}, \mathbf{b}, \mathbf{c}$ as coterminous edges is 61 cubic units, then the number of possible values of $\lambda$ is

If two vectors $\mathbf{a}$ and $\mathbf{b}$ which are perpendicular to each other are such that $|\mathbf{a}|=8$ and $|\mathbf{b}|=3$, then $|\mathbf{a}-2 b|=$

Let $\mathbf{a}$ and $\mathbf{b}$ be non-collinear vectors. If the vectors $(\lambda-1) \mathbf{a}+2 \mathbf{b}$ and $3 \mathbf{a}+\lambda \mathbf{b}$ are collinear, then the set of all possible values of $\lambda$ is