If the vectors $(2 \hat{\imath} - q \hat{\jmath} + 3 \hat{k})$ and $(4 \hat{\imath} - 5 \hat{\jmath} + 6 \hat{k})$ are collinear,then the value of $q$ is

If $\vec{a}, \vec{b}, \vec{c}$ are position vectors of points $A, B, C$ respectively,with $2 \vec{a}+3 \vec{b}-5 \vec{c}=\vec{0}$,then the ratio in which point $C$ divides segment $AB$ is

The triad $(x, y, z)$ of real numbers such that $(3 \hat{i}-\hat{j}+2 \hat{k})=(2 \hat{i}+3 \hat{j}-\hat{k}) x+(\hat{i}-2 \hat{j}+2 \hat{k}) y+(-2 \hat{i}+\hat{j}-2 \hat{k}) z$ is

If the orthocentre of the triangle whose vertices are $2 \hat{i}+3 \hat{j}+5 \hat{k}$,$5 \hat{i}+2 \hat{j}+3 \hat{k}$,and $3 \hat{i}+5 \hat{j}+2 \hat{k}$ is $x \hat{i}+y \hat{j}+z \hat{k}$,then:

If the vectors $3\,i + 2\,j - k$ and $6\,i - 4xj + yk$ are parallel,then the value of $x$ and $y$ will be

If the points $a + b$,$a - b$,and $a + kb$ are collinear,then $k =$