Practice section 2 of 9

Given that $\mathbf a$ and $\mathbf b$ are the position vectors of $A$ and $B$ respectively relative to origin $O$, find, in terms of $\mathbf a,\mathbf b$, the position vectors of: (i) $P$ which divides $AB$ in the ratio $3:7$; (ii) $R$ on $BA$ produced such that $AR/BR=2/7$; (iii) $Q$ on $AB$ produced such that $AQ/BQ=7/3$.

Given the line segment $AB$, where $A(3,1,2)$ and $B(2,3,5)$, find the coordinates of $P$ on (i) $AB$ with $AP:PB=2:3$; (ii) $AB$ produced with $AP:PB=2:1$.

Nonzero, nonparallel vectors $\mathbf a,\mathbf b$ are the position vectors of $A,B$ relative to $O$. $P$ is the midpoint of $AB$ and $Q$ is the midpoint of $OP$. The line $AQ$ produced cuts $OB$ at $R$. Find, in terms of $\mathbf a,\mathbf b$, (i) the position vector of $Q$ relative to $O$ and (ii) $\overrightarrow{AQ}$. Prove that $\overrightarrow{OR}=\tfrac13\mathbf b$ and find $AQ:QR$.

Self-Reading: A Special Point of a Triangle — The Centroid. Choose one vertex of a triangle as origin $O$, with the other vertices $A,B$.

A median joins a vertex to the midpoint of the opposite side; $AM$ is the median from $A$. Let $G$ be the intersection of the medians from $A$ and $B$.

(i) Express the position vector of $G$ in terms of the position vectors of $A,B$.

(ii) Determine $AG:GM$.

(iii) Show that $G$ also lies on the median from $O$.

Given $\mathbf a=\begin{pmatrix}1\\2\\1\end{pmatrix}$ and $\mathbf b=\begin{pmatrix}2\\3\\-1\end{pmatrix}$, find $(3\mathbf a-\mathbf b)\cdot(2\mathbf a+\mathbf b)$.

If $\mathbf a=\mathbf i+4\mathbf j-\mathbf k$, $\mathbf b=\mathbf i-\mathbf j+3\mathbf k$ and $\mathbf c=2\mathbf i+\mathbf j-5\mathbf k$, find (i) $\mathbf a\cdot\mathbf c$ and (ii) $(2\mathbf a+3\mathbf b)\cdot\mathbf c$.