In $\Delta ABC$,if $A(2, 4, -1)$,$B(4, 5, 1)$,and $C(3, 6, -3)$,then $\Delta ABC$ is a ... triangle.

Let $ABC$ be a triangle whose circumcentre is at $P$. If the position vectors of $A, B, C$ and $P$ are $\vec{a}, \vec{b}, \vec{c}$ and $\frac{\vec{a} + \vec{b} + \vec{c}}{4}$ respectively,then the position vector of the orthocentre of this triangle is:

Let $\vec{a}$ and $\vec{b}$ be two non-collinear vectors. For what values of $x$ and $y$ is the equation $2\vec{u} - \vec{v} = \vec{w}$ true,where $\vec{u} = x\vec{a} + 2y\vec{b}$,$\vec{v} = -2y\vec{a} + 3x\vec{b}$,and $\vec{w} = 4\vec{a} - 2\vec{b}$?

If the magnitude of the sum of two unit vectors is greater than the magnitude of their difference and less than $\sqrt{3}$ times the magnitude of their difference,then the complete set of values for the angle $\theta$ between the vectors is

Let $G$ be the centroid of a triangle $ABC$ and $O$ be any other point in that plane,then $\overline{OA}+\overline{OB}+\overline{OC}+\overline{OG}=$

Let $\vec{a}=x \hat{i}+y \hat{j}+z \hat{k}$ and $x=2 y$. If $|\vec{a}|=5 \sqrt{2}$ and $\vec{a}$ makes an angle of $135^{\circ}$ with the $z$-axis,then $\vec{a}=$