If $\vec{u} = \hat{j} + 4\hat{k}$,$\vec{v} = \hat{i} + 3\hat{k}$ and $\vec{w} = \cos \theta \hat{i} + \sin \theta \hat{j}$ are vectors in $3$-dimensional space,then the maximum possible value of $|(\vec{u} \times \vec{v}) \cdot \vec{w}|$ is

$a \cdot (b \times c)$ is equal to

If the vectors $2\hat{i}-\hat{j}+3\hat{k}$,$\hat{i}+4\hat{j}+\hat{k}$,and $4\hat{i}+p\hat{j}+\hat{k}$ are coplanar,then $p=$

Let $\overrightarrow{a}=\hat{i}-2 \hat{j}+3 \hat{k}$,$\overrightarrow{b}=2 \hat{i}+3 \hat{j}-\hat{k}$ and $\overrightarrow{c}=\lambda \hat{i}+\hat{j}+(2 \lambda-1) \hat{k}$. If $\overrightarrow{c}$ is parallel to the plane containing $\overrightarrow{a}$ and $\overrightarrow{b}$,then $\lambda$ is equal to

If $\bar{a}=\hat{i}+\hat{j}+\hat{k}$,$\bar{b}=4\hat{i}+3\hat{j}+4\hat{k}$,and $\bar{c}=\hat{i}+\alpha\hat{j}+\beta\hat{k}$ are linearly dependent vectors and $|\bar{c}|=\sqrt{3}$,then the values of $\alpha$ and $\beta$ are respectively.

$(a+b) \cdot(b+c) \times(a+b+c)$ is equal to