Let $\vec{u}, \vec{v}, \vec{w}$ be vectors such that $\vec{u} + \vec{v} + \vec{w} = \vec{0}$. If $|\vec{u}| = 3$,$|\vec{v}| = 4$,and $|\vec{w}| = 5$,then $\vec{u} \cdot \vec{v} + \vec{v} \cdot \vec{w} + \vec{w} \cdot \vec{u}$ is:

Let $ABC$ be a triangle and $\bar{a}, \bar{b}, \bar{c}$ be the position vectors of $A, B, C$ respectively. Let $D$ divide $BC$ in the ratio $3:1$ internally and $E$ divide $AD$ in the ratio $4:1$ internally. Let $BE$ meet $AC$ in $F$. If $E$ divides $BF$ in the ratio $3:2$ internally, then the position vector of $F$ is

If $a = \hat{i} + 4 \hat{j}$, $b = 2 \hat{i} - 2 \hat{j}$, and $c = 5 \hat{i} + 9 \hat{j}$, then $c$ is equal to:

The vectors $\overline{AB} = 3\hat{i} + 4\hat{k}$ and $\overline{AC} = 5\hat{i} - 2\hat{j} + 4\hat{k}$ represent the sides of a triangle $ABC$. What is the length of the median passing through vertex $A$?

Five points given by $A, B, C, D, E$ are in a plane. Three forces $\overrightarrow{AC}, \overrightarrow{AD},$ and $\overrightarrow{AE}$ act at $A$,and three forces $\overrightarrow{CB}, \overrightarrow{DB},$ and $\overrightarrow{EB}$ act at $B$. Then their resultant is:

If $\vec{a}$ and $\vec{b}$ are two collinear vectors,then which of the following is incorrect?