The magnitude of the sum of vectors $(1, -\sqrt{2})$ and $(2, \sqrt{2})$ is $\ldots$

If the position vectors of points $P, Q, R,$ and $S$ are $2\hat{i} + 3\hat{j} + 5\hat{k}$,$\hat{i} + 2\hat{j} + 3\hat{k}$,$-5\hat{i} + 4\hat{j} - 2\hat{k}$,and $\hat{i} + 10\hat{j} + 10\hat{k}$ respectively,then:

If $\vec{a}=\hat{i}+2 \hat{j}-3 \hat{k}$ and $\vec{b}=2 \hat{i}-3 \hat{j}-5 \hat{k}$,then

Let $a = i$ be a vector which makes an angle of $120^\circ$ with a unit vector $b$. Then the unit vector $(a + b)$ is

$A$ vector $\vec{r}$ is inclined at equal angles to the three axes. If the magnitude of $\vec{r}$ is $2 \sqrt{3}$ units,find $\vec{r}$.

If $\vec{a} = \hat{i}+\hat{j}-\hat{k}$ and $\vec{b} = 2\hat{i}-3\hat{j}+\hat{k}$ are adjacent sides of a parallelogram,then the lengths of its diagonals are