If $\hat{i}, \hat{j}, \hat{k}$ are unit vectors,then which of the following is true?

If three points $A$,$B$,and $C$ have position vectors $(1, x, 3)$,$(3, 4, 7)$,and $(y, -2, -5)$ respectively and if they are collinear,then $(x, y)$ is

Consider two points $P$ and $Q$ with position vectors $\overrightarrow{OP} = 3\vec{a} - 2\vec{b}$ and $\overrightarrow{OQ} = \vec{a} + \vec{b}$. Find the position vector of a point $R$ which divides the line segment joining $P$ and $Q$ in the ratio $2:1$ internally.

If $\vec{a}=\hat{i}+\hat{j}+2 \hat{k}$ and $\vec{b}=2 \hat{i}+\hat{j}+2 \hat{k},$ find the unit vector in the direction of
$(i)$ $6 \vec{b}$
(ii) $2 \vec{a}-\vec{b}$

Let $u$ and $v$ be two vectors in a plane. Then any vector $w$ in the plane can be written as $w = au + bv$ for some scalars $a$ and $b$ if and only if

Find the scalar and vector components of the vector with initial point $(2,1)$ and terminal point $(-5,7).$