The perimeter of a triangle with sides represented by vectors $\vec{a} = 3\hat{i} + 4\hat{j} + 5\hat{k}$,$\vec{b} = 4\hat{i} - 3\hat{j} - 5\hat{k}$,and $\vec{c} = 7\hat{i} + \hat{j}$ is:

Let $H$ be the orthocentre of an acute angled $\triangle ABC$ and $O$ be its circumcenter. Then,$\vec{HA} + \vec{HB} + \vec{HC}$

If $\bar{a} = (x + 2y - 3) \bar{i} + (2x - y + 3) \bar{j}$ and $\bar{b} = (3x - 2y) \bar{i} + (x - y + 1) \bar{j}$ are two vectors such that $\bar{a} = 2 \bar{b}$, then $y - 5x =$

In a right-angled triangle,if the position vector of the vertex having the right angle is $\vec{A} = -3\hat{i} + 5\hat{j} + 2\hat{k}$ and the position vector of the midpoint of its hypotenuse is $\vec{M} = 6\hat{i} + 2\hat{j} + 5\hat{k}$,then the position vector of its centroid is

Find the unit vector in the direction of the sum of vectors $\vec{a} = 2\hat{i} - \hat{j} + \hat{k}$ and $\vec{b} = 2\hat{j} + \hat{k}$.

Find the centroid of the triangle whose vertices are $i + 2j, 2i + j, i + j + k$.