If $\overrightarrow{AB} = (2\hat{i} + 3\hat{j} - \hat{k})$ and $A(1, 2, -1)$ is the given point,then the coordinates of $B$ are

If $\overrightarrow{AB} = 2\hat{i} + 3\hat{j} - 6\hat{k}$ and $\overrightarrow{BC} = 6\hat{i} - 2\hat{j} + 3\hat{k}$ are the vectors along two sides of a triangle $ABC$,then the perimeter of triangle $ABC$ is:

Find a vector in the direction of vector $\vec{a} = \hat{i} - 2\hat{j}$ that has magnitude $7$ units.

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:

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 $\vec{a}$.

If $|\overline{a}|=2, |\overline{b}|=3, |\overline{c}|=5$ and each of the angles between the vectors $\overline{a}$ and $\overline{b}$,$\overline{b}$ and $\overline{c}$,and $\overline{c}$ and $\overline{a}$ is $60^{\circ}$,then the value of $|\overline{a}+\overline{b}+\overline{c}|$ is