$A$ zero vector has

Let $A, B$ and $C$ be three points on a circle of radius $R$. If $O$ is the centre of the circle and $\angle AOB = 45^{\circ}, \angle BOC = 45^{\circ}$,then the resultant of $\vec{OA}, \vec{OB}$ and $\vec{OC}$ has magnitude

If the points with position vectors $(\alpha \hat{i}+10 \hat{j}+13 \hat{k})$,$(6 \hat{i}+11 \hat{j}+11 \hat{k})$,and $(\frac{9}{2} \hat{i}+\beta \hat{j}-8 \hat{k})$ are collinear,then $(19 \alpha-6 \beta)^2=$

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 $\bar{a}, \bar{b}, \bar{c}$ are three non-zero vectors,no two of them are collinear,$\bar{a}+2 \bar{b}$ is collinear with $\bar{c}$,and $\bar{b}+3 \bar{c}$ is collinear with $\bar{a}$,then $\bar{a}+2 \bar{b}$ is equal to:

Which of the following is not a unit vector for all values of $\theta$?