$a$ and $b$ are non-collinear vectors. If $c=(x-2)a+b$ and $d=(2x+1)a-b$ are collinear vectors,then the value of $x = \ldots$

If $a$ and $b$ represent two non-collinear vectors, the equation $r = ta + (1-t)b$ represents

If the coordinates of points $A, B, C,$ and $D$ are $(1, 2, 3), (4, 5, 7), (-4, 3, -6),$ and $(2, 9, 2)$ respectively,then the angle between $AB$ and $CD$ is:

$A$ point $C$ with position vector $\frac{3 \bar{a}+4 \bar{b}-5 \bar{c}}{3}$ (where $\bar{a}, \bar{b}$ and $\bar{c}$ are non-coplanar vectors) divides the line segment joining $A$ and $B$ in the ratio $2:1$. If the position vector of $A$ is $\bar{a}-2 \bar{b}+3 \bar{c}$,then find the position vector of $B$.

If $\vec{a}=\hat{i}+2 \hat{j}-3 \hat{k}$ and $\vec{b}=2 \hat{i}-3 \hat{j}-5 \hat{k}$,then

The direction cosines of the vector $\vec{a} = -2 \hat{i} + \hat{j} - 5 \hat{k}$ are