The system of unit vectors $i, j, k$ is

Find the values of $x$ and $y$ so that the vectors $2 \hat{i} + 3 \hat{j}$ and $x \hat{i} + y \hat{j}$ are equal.

If $\vec{PO} + \vec{OQ} = \vec{QO} + \vec{OR}$,then

If the position vectors of the points $A, B, C, D$ are $2i + 3j + 5k, i + 2j + 3k, -5i + 4j - 2k$ and $i + 10j + 10k$ respectively,then:

The position vectors of the points $P$ and $Q$ are respectively $-2 \bar{i}-3 \bar{j}+\bar{k}$ and $3 \bar{i}+3 \bar{j}+2 \bar{k}$. The ratio in which the point having position vector $\frac{-9}{2} \bar{i}-6 \bar{j}+\frac{1}{2} \bar{k}$ divides the line segment joining $P$ and $Q$ is

If $a = \alpha \hat{i} + 3 \hat{j} - 6 \hat{k}$ and $b = 2 \hat{i} - \hat{j} + \beta \hat{k}$,then the values of $\alpha, \beta$ so that $a$ and $b$ may be collinear are