Let the vectors $\overrightarrow{AB} = 2\hat{i} + 2\hat{j} + \hat{k}$ and $\overrightarrow{AC} = 2\hat{i} + 4\hat{j} + 4\hat{k}$ be two sides of a triangle $ABC$. If $G$ is the centroid of $\triangle ABC$, then $\frac{27}{7}(\overrightarrow{AG})^2 + 5 =$

Let $\overrightarrow{OA}=\hat{i}+2 \hat{j}-2 \hat{k}$ and $\overrightarrow{OB}=-2 \hat{i}-3 \hat{j}+6 \hat{k}$ be the position vectors of two points $A$ and $B$. If $C$ is a point on the bisector of $\angle AOB$ and $OC=\sqrt{42}$,then $\overrightarrow{OC}=$

For any vector $\vec{a} = a_1 \hat{i} + a_2 \hat{j} + a_3 \hat{k}$,with $10|a_i| < 1$,$i = 1, 2, 3$,consider the following statements:
$(A): \max \{|a_1|, |a_2|, |a_3|\} \leq |\vec{a}|$
$(B): |\vec{a}| \leq 3 \max \{|a_1|, |a_2|, |a_3|\}$

$R$ divides the line joining two points $P$ and $Q$ whose position vectors are $\hat{i}+2 \hat{j}-\hat{k}$ and $-\hat{i}+\hat{j}+\hat{k}$ respectively in the ratio $2: 1$ externally. $S$ divides $PQ$ internally in the ratio $2: 1$. Then,the position vector of the midpoint of the line joining $R$ and $S$ is

Let $A$ and $B$ be points with position vectors $a$ and $b$ with respect to the origin $O$. If the point $C$ on $OA$ is such that $2AC = CO$,$CD$ is parallel to $OB$ and $|\overrightarrow{CD}| = 3|\overrightarrow{OB}|$,then $\overrightarrow{AD}$ is equal to

Classify the following as scalar or vector quantity:
distance