In a triangle $ABC$,if $2\overrightarrow{AC} = 3\overrightarrow{CB}$,then $2\overrightarrow{OA} + 3\overrightarrow{OB}$ equals

If vectors $a$ and $b$ represent two consecutive sides of a regular hexagon,then the vectors representing the remaining four sides in order are:

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 the vector $\vec{a} = \hat{i} + \hat{j} + \hat{k}$ makes angles $\alpha, \beta, \gamma$ with vectors $\hat{i}, \hat{j}, \hat{k}$ respectively,then:

If $\alpha, \beta, \gamma$ are real numbers such that $(\frac{7}{3}+\beta) \hat{i}-\hat{j}+(\alpha+\gamma) \hat{k}=\frac{5}{3}(\alpha \hat{i}+\hat{j}-\hat{k})+\beta(2 \hat{j}+\hat{k})+(\hat{i}+\gamma \hat{j}+3 \hat{k})$, then $5 \alpha-9 \beta+13 \gamma=$

If $p = 7i - 2j + 3k$ and $q = 3i + j + 5k,$ then the magnitude of $p - 2q$ is