Let $\vec{a}$ and $\vec{b}$ be non-collinear vectors. If the vectors $(\lambda-1) \vec{a}+2 \vec{b}$ and $3 \vec{a}+\lambda \vec{b}$ are collinear,then the set of all possible values of $\lambda$ is

Suppose that $\bar{p}, \bar{q}$ and $\bar{r}$ are three non-coplanar vectors in $\mathbb{R}^3$. Let the components of a vector $\bar{s}$ along $\bar{p}, \bar{q}$ and $\bar{r}$ be $4, 3$ and $5$ respectively. If the components of this vector $\bar{s}$ along $(-\bar{p}+\bar{q}+\bar{r}), (\bar{p}-\bar{q}+\bar{r})$ and $(-\bar{p}-\bar{q}+\bar{r})$ are $x, y$ and $z$ respectively,then the value of $2x+y+z$ is

If the vector $\vec{a} = \hat{i} + \hat{j} + \hat{k}$ makes angles $\alpha, \beta, \gamma$ with vectors $\hat{i}, \hat{j}, \hat{k}$ respectively,then:

Find the scalar components and magnitude of the vector joining the points $P(x_{1}, y_{1}, z_{1})$ and $Q(x_{2}, y_{2}, z_{2}).$

In a regular hexagon $ABCDEF$,$\overrightarrow{AE} = $

The sum of the lengths of projections of $p \hat{i} + q \hat{j} + r \hat{k}$ on the coordinate axes, where $p = 4, q = -5, r = 7$, is: (in $\text{ units}$)