Let $\vec{u}$ and $\vec{v}$ be two non-zero vectors with the intermediate angle $45^{\circ}$. Then $|\vec{u} \times \vec{v}|=$

Let the position vectors of the vertices $A, B$ and $C$ of a tetrahedron $ABCD$ be $\hat{i}+2\hat{j}+\hat{k}$,$\hat{i}+3\hat{j}-2\hat{k}$ and $2\hat{i}+\hat{j}-\hat{k}$ respectively. The altitude from the vertex $D$ to the opposite face $ABC$ meets the median line segment through $A$ of the triangle $ABC$ at the point $E$. If the length of $AD$ is $\frac{\sqrt{110}}{3}$ and the volume of the tetrahedron is $\frac{\sqrt{805}}{6\sqrt{2}}$,then the position vector of $E$ is

If $a, b, c$ are three non-zero,non-coplanar vectors and $b_1 = b - \frac{b \cdot a}{|a|^2} a$,$b_2 = b + \frac{b \cdot a}{|a|^2} a$,$c_2 = c - \frac{c \cdot a}{|a|^2} a - \frac{c \cdot b_1}{|b_1|^2} b_1$,$c_3 = c - \frac{c \cdot a}{|a|^2} a - \frac{c \cdot b_2}{|b_2|^2} b_2$,and $c_4 = a - \frac{c \cdot a}{|a|^2} a$. Then which of the following is a set of mutually orthogonal vectors?

If $a = 2i + 2j + 3k$,$b = -i + 2j + k$,and $c = 3i + j$,then $a + tb$ is perpendicular to $c$ if $t = $

Let $PQR$ be a triangle such that $\overrightarrow{PQ}=-2\hat{i}-\hat{j}+2\hat{k}$ and $\overrightarrow{PR}=a\hat{i}+b\hat{j}-4\hat{k}$, where $a, b \in \mathbb{Z}$. Let $S$ be the point on $QR$, which is equidistant from the lines $PQ$ and $PR$. If $|\overrightarrow{PR}|=9$ and $\overrightarrow{PS}=\hat{i}-7\hat{j}+2\hat{k}$, then the value of $3a-4b$ is . . . . . . .

If the vectors $\hat{i}-2x\hat{j}-3y\hat{k}$ and $\hat{i}+3x\hat{j}+2y\hat{k}$ are orthogonal to each other,then the locus of the point $(x, y)$ is