Find the values of $x, y$ and $z$ so that the vectors $\vec{a} = x \hat{i} + 2 \hat{j} + z \hat{k}$ and $\vec{b} = 2 \hat{i} + y \hat{j} + \hat{k}$ are equal.

In $\triangle ABC$,$L, M, N$ are points on $BC, CA, AB$ respectively,dividing them in the ratio $1:2, 2:3, 3:5$. If the point $K$ divides $AB$ in the ratio $5:3$,then find the value of $\left| \frac{\vec{AL} + \vec{BM} + \vec{CN}}{\vec{CK}} \right|$.

If non-zero vectors $\vec{a}$,$\vec{b}$,and $\vec{c}$ are related by $\vec{a} = 8\vec{b}$ and $\vec{c} = -7\vec{b}$,find the angle between $\vec{a}$ and $\vec{c}$.

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

Find the position vector of a point $R$ which divides the line joining two points $P$ and $Q$ whose position vectors are $(2 \vec{a}+\vec{b})$ and $(\vec{a}-3 \vec{b})$ externally in the ratio $1:2$. Also,show that $P$ is the mid-point of the line segment $RQ$.

If $A(1, 0, 0)$,$B(0, 1, 0)$,and $C(0, 0, 1)$ are given,and $\vec{AB} = \vec{CX}$,then the point $X$ is: