If the vectors $\vec{a} = \lambda \hat{i} + 2\hat{j} - 3\hat{k}$ and $\vec{b} = \sqrt{\lambda} \hat{i} + \sqrt{13} \hat{j}$ have the same magnitude,then the value of $\lambda$ is:

If $a = 2i + j - 8k$ and $b = i + 3j - 4k,$ then the magnitude of $a + b = $

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 $\hat{i}-2 \hat{j}+3 \hat{k}$,$2 \hat{i}+3 \hat{j}+\hat{k}$,and $-3 \hat{i}-\hat{j}-2 \hat{k}$ are the position vectors of three points $A$,$B$,and $C$ respectively,then $A$,$B$,and $C$

Which of the following statements is true?

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 $\overrightarrow{r}=$