If $a, b$ and $c$ are mutually perpendicular vectors of the same magnitude,then the cosine of the angle between $a$ and $a+b+c$ is

Let $a = 2i + j + k$,$b = i + 2j - k$,and a unit vector $c$ be coplanar. If $c$ is perpendicular to $a$,then $c = \dots$

If $3\vec{a} - 5\vec{b}$ and $2\vec{a} + \vec{b}$ are perpendicular to each other,and $\vec{a} + 4\vec{b}$ and $-\vec{a} + \vec{b}$ are also perpendicular to each other,and $\theta$ is the angle between $\vec{a}$ and $\vec{b}$,then find $\cos \theta$.

If the position vectors of the vertices $A, B$ and $C$ of a $\Delta ABC$ are respectively $4\hat{i} + 7\hat{j} + 8\hat{k}$,$2\hat{i} + 3\hat{j} + 4\hat{k}$ and $2\hat{i} + 5\hat{j} + 7\hat{k}$,then the position vector of the point,where the bisector of $\angle A$ meets $BC$ is

$a, b$ and $c$ are three vectors such that $|a|=1, |b|=2, |c|=3$ and $b, c$ are perpendicular. If the projection of $b$ on $a$ is the same as the projection of $c$ on $a$, then $|a-b+c|$ is equal to

If $a, b$ and $c$ are three vectors such that $|a|=3, |b|=4$ and $|c|=5$ and $a+b+c=0$,then $a \cdot b$ is equal to