Let $\vec{a} = -\hat{i} - \hat{k}$,$\vec{b} = -\hat{i} + \hat{j}$,and $\vec{c} = \hat{i} + 2\hat{j} + 3\hat{k}$ be three given vectors. If $\vec{r}$ is a vector such that $\vec{r} \times \vec{b} = \vec{c} \times \vec{b}$ and $\vec{r} \cdot \vec{a} = 0$,then the value of $\vec{r} \cdot \vec{b}$ is

The projection of vector $\vec{a} = \hat{i} + 2\hat{j} + \hat{k}$ on vector $\vec{b} = 2\hat{i} + 3\hat{j} + 2\hat{k}$ is . . . . . . .

Let $\vec{\alpha}, \vec{\beta}, \vec{\gamma}$ be three non-zero vectors which are pairwise non-collinear. If $\vec{\alpha}+3 \vec{\beta}$ is collinear with $\vec{\gamma}$ and $\vec{\beta}+2 \vec{\gamma}$ is collinear with $\vec{\alpha}$,then $\vec{\alpha}+3 \vec{\beta}+6 \vec{\gamma}$ is

Three vectors of magnitudes $a, 2a, 3a$ are along the directions of the diagonals of $3$ adjacent faces of a cube that meet at a point. The magnitude of the sum of these vectors is: (in $a$)

An arc $PQ$ of a circle subtends a right angle at its centre $O$. The midpoint of the arc $PQ$ is $R$. If $\vec{OP}=\vec{u}$,$\vec{OR}=\vec{v}$ and $\vec{OQ}=\alpha \vec{u}+\beta \vec{v}$,then $\alpha, \beta^2$ are the roots of the equation

Let $\vec{p}$ and $\vec{q}$ be the position vectors of points $P$ and $Q$ respectively,with respect to the origin $O$,and let $|\vec{p}|=p, |\vec{q}|=q$. The points $R$ and $S$ divide the line segment $PQ$ internally and externally in the ratio $2:3$ respectively. If $\vec{OR}$ and $\vec{OS}$ are perpendicular,then: