If $\bar{a}, \bar{b}, \bar{c}$ are non-coplanar vectors and the points represented by position vectors $\bar{a}-2 \bar{b}+3 \bar{c}$,$-4 \bar{a}+5 \bar{b}-6 \bar{c}$,and $x \bar{a}-9 \bar{b}+z \bar{c}$ are collinear,then $2x-z=$

If $2 \hat{i}-\hat{j}+\hat{k}$ and $\hat{i}-3 \hat{j}-5 \hat{k}$ are the position vectors of the points $A$ and $B$ respectively,$C$ divides $AB$ in the ratio $2:3$ and $M$ is the mid-point of $AB$,then $5(\text{position vector of } C) - 2(\text{position vector of } M) =$

Let $a = \hat{i} + x \hat{j} + \hat{k}$,$b = \hat{i} + \hat{j} + \hat{k}$ and $|a + b| = |a| + |b|$,then

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 $\overline{r} = -4 \hat{i} - 6 \hat{j} - 2 \hat{k}$ is a linear combination of the vectors $\overline{a} = -\hat{i} - 4 \hat{j} + 3 \hat{k}$ and $\overline{b} = -8 \hat{i} - \hat{j} + 3 \hat{k}$,then which of the following is true?

If $\vec{a}, \vec{b}, \vec{c}$ are $3$ vectors such that $|\vec{a}|=5, |\vec{b}|=8, |\vec{c}|=11$ and $\vec{a}+\vec{b}+\vec{c}=\overrightarrow{0}$,then the angle between the vectors $\vec{a}$ and $\vec{b}$ is