If $a=\hat{i}+\hat{j}+\hat{k}$, $a \cdot b=1$ and $a \times b=\hat{j}-\hat{k}$, then $b=$

If $\vec{a}, \vec{b}$ and $\vec{c}$ are unit vectors such that $\vec{a}$ is perpendicular to $\vec{b}$ and $\vec{c}$,and $|\vec{a} + \vec{b} + \vec{c}| = 1$,then the angle between $\vec{b}$ and $\vec{c}$ is:

Let $\bar{a}, \bar{b}$ and $\bar{c}$ be vectors of magnitude $2, 3$ and $4$ respectively. If $\bar{a}$ is perpendicular to $(\bar{b}+\bar{c})$,$\bar{b}$ is perpendicular to $(\bar{c}+\bar{a})$ and $\bar{c}$ is perpendicular to $(\bar{a}+\bar{b})$,then the magnitude of $\bar{a}+\bar{b}+\bar{c}$ is

Let $\vec a, \vec b, \vec c$ be three vectors such that $\vec a \perp (\vec b + \vec c)$,$\vec b \perp (\vec c + \vec a)$,and $\vec c \perp (\vec a + \vec b)$. If $|\vec a| = 1, |\vec b| = 2, |\vec c| = 3$,then $|\vec a + \vec b + \vec c| = \dots$

If $\overline{a}, \overline{b}, \overline{c}$ are three vectors,$|\overline{a}|=2, |\overline{b}|=4, |\overline{c}|=1$,$|\overline{b} \times \overline{c}|=\sqrt{15}$ and $\overline{b}=2 \overline{c}+\lambda \overline{a}$,then the value of $\lambda$ is

Let $P$ be a point in the plane of the vectors $\overrightarrow{AB}=3\hat{i}+\hat{j}-\hat{k}$ and $\overrightarrow{AC}=\hat{i}-\hat{j}+3\hat{k}$ such that $P$ is equidistant from the lines $AB$ and $AC$. If $|\overrightarrow{AP}|=\frac{\sqrt{5}}{2}$, then the area of the triangle $ABP$ is: