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

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

The ratio in which $\hat{i}+2 \hat{j}+3 \hat{k}$ divides the join of $-2 \hat{i}+3 \hat{j}+5 \hat{k}$ and $7 \hat{i}-\hat{k}$ is

If a vector $a$ of magnitude $50$ is collinear with the vector $b = 6i - 8j - \frac{15}{2}k$, and makes an acute angle with the positive direction of the $z$-axis, then the vector $a$ is equal to:

The vertices of triangle $ABC$ are $A \equiv (3, 0, 0)$,$B \equiv (0, 0, 4)$,and $C \equiv (0, 5, 4)$. Find the position vector of the point $D$ where the angle bisector of $\angle A$ meets $BC$.

If $(\alpha, \beta, \gamma)$ is a triad of real numbers satisfying $\hat{i}-2 \hat{j}+5 \hat{k}=\alpha(\hat{i}+\hat{j}+\hat{k})+\beta(\hat{i}+2 \hat{j}+3 \hat{k})+\gamma(2 \hat{i}-\hat{j}+\hat{k}),$ then $\alpha^2-\beta^2+\gamma^2=$