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 $(19 \alpha-6 \beta)^2=$

Three non-zero non-collinear vectors $\vec{a}, \vec{b}$ and $\vec{c}$ are such that $\vec{a}+3\vec{b}$ is collinear with $\vec{c}$, and $3\vec{b}+2\vec{c}$ is collinear with $\vec{a}$. Then $\vec{a}+3\vec{b}+2\vec{c}$ equals to

If $\vec{a}$ is a non-zero vector of modulus $|\vec{a}|$ and $m$ is a non-zero scalar,then $m\vec{a}$ is a unit vector if:

Let $\vec{a}=2 \hat{i}+\hat{j}-\hat{k}$ and $\vec{b}=\hat{i}+3 \hat{j}-5 \hat{k}$ be two vectors,and $\overrightarrow{r}$ be a vector along the vector $3 \overrightarrow{a}-2 \overrightarrow{b}$ such that $|\overrightarrow{r}|=\sqrt{74}$. If the direction of $\vec{r}$ is opposite to that of $3 \vec{a}-2 \vec{b}$,then $\overrightarrow{r}=$

If $\vec{a}$ and $\vec{b}$ are non-zero vectors which are linearly dependent such that $\frac{|\vec{a} + \vec{b}|}{|\vec{a} - \vec{b}|} = 2$ and $|\vec{b}| > |\vec{a}|$,then:

Let $H$ be the orthocentre of an acute angled $\triangle ABC$ and $O$ be its circumcenter. Then,$\vec{HA} + \vec{HB} + \vec{HC}$