If three points $A, B, C$ are collinear,whose position vectors are $i - 2j - 8k$,$5i - 2k$,and $11i + 3j + 7k$ respectively,then the ratio in which $B$ divides $AC$ is

Vectors $\vec{a}$,$\vec{b}$,and $\vec{c}$ are of the same length and they make equal angles with each other when taken in pairs. If $\vec{a} = \hat{i} - \hat{j}$,$\vec{b} = \hat{j} + \hat{k}$,and $\vec{c}$ makes an obtuse angle with the $x$-axis,find the vector $\vec{c}$.

If the position vectors of two points $A$ and $B$ are $\vec{a} + 3\vec{b}$ and $\vec{a} - 2\vec{b}$ respectively,find the position vector of the point that divides $AB$ in the ratio $2:5$.

Answer the following as true or false.
Two collinear vectors are always equal in magnitude.

Let $\vec{i}+\vec{j}+\vec{k}$,$a_1 \vec{i}+b_1 \vec{j}+c_1 \vec{k}$,$a_2 \vec{i}+b_2 \vec{j}+c_2 \vec{k}$,and $a_3 \vec{i}+b_3 \vec{j}+c_3 \vec{k}$ be the position vectors of the points $A, B, C, D$ respectively. The position vector of the centroid of the triangular face $BCD$ is $\frac{2}{3}(\vec{i}+\vec{j}+\vec{k})$. If $\alpha \vec{i}+\beta \vec{j}+\gamma \vec{k}$ is the position vector of the centroid of the tetrahedron $ABCD$,then find the value of $2 \alpha+\beta+\gamma$.

If $A(2, 3, 5)$,$B(1, 2, 3)$,$C(-5, 4, -2)$,and $D(1, 10, 10)$,then ...