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:

Let $\vec{v}$ be a vector in the plane such that $|\vec{v} - \hat{i}| = |\vec{v} - 2\hat{j}| = |\vec{v} - \hat{j}|$. Then,$|\vec{v}|$ lies in the interval

Let $u, v, w$ be vectors such that $u + v + w = 0$. If $|u| = 3, |v| = 4,$ and $|w| = 5,$ find the value of $u \cdot v + v \cdot w + w \cdot u$.

If $\hat{i}-2 \hat{j}+3 \hat{k}$,$2 \hat{i}+3 \hat{j}+\hat{k}$,and $-3 \hat{i}-\hat{j}-2 \hat{k}$ are the position vectors of three points $A$,$B$,and $C$ respectively,then $A$,$B$,and $C$

$(a \cdot i)i + (a \cdot j)j + (a \cdot k)k = $

If the sides of a regular hexagon $ABCDEF$ are given by $\vec{AB} = \bar{a}$ and $\vec{BC} = \bar{b}$,then $\vec{FA} = .....$