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

Let $A, B$ and $C$ be three points on a circle of radius $R$. If $O$ is the centre of the circle and $\angle AOB = 45^{\circ}, \angle BOC = 45^{\circ}$,then the resultant of $\vec{OA}, \vec{OB}$ and $\vec{OC}$ has magnitude

If $\vec{a} = \hat{i}+\hat{j}-\hat{k}$ and $\vec{b} = 2\hat{i}-3\hat{j}+\hat{k}$ are adjacent sides of a parallelogram,then the lengths of its diagonals are

If a unit vector $\vec{a}$ makes angles $\frac{\pi}{3}$ with $\hat{i}$,$\frac{\pi}{4}$ with $\hat{j}$ and an acute angle $\theta$ with $\hat{k}$,then find $\theta$ and the components of $\vec{a}$.

Let $A$ be the set of vectors $a = (a_1, a_2, a_3)$ satisfying $\left(\sum_{i=1}^3 \frac{a_i}{2^i}\right)^2 = \sum_{i=1}^3 \frac{a_i^2}{2^i}$. Then,

Answer the following as true or false.
Two collinear vectors are always equal in magnitude.